Download Chapter 4, Mathematics

What is Philosophy Chapter 4
by Richard Thompson
Mathematics
(last edited on 25th April 2012)
This chapter assumes familiarity with formal logic, described in Chapter 2 of these
notes
In the Greek world the only successful field of theoretical knowledge, apart from
Aristotle’s formal Logic, was Mathematics and the most highly developed branch of
Mathematics was Geometry.
Chinese and Indian Mathematicians independently developed arithmetic and the
rudiments of algebra. The Eastern and Western traditions appear to have been quite
separate until the Arabs brought Indian arithmetic to Europe around the thirteenth
century. Until then Europeans did not know the decimal system of numbering and had no
efficient procedure for calculation, depending on the abacus for all but the very simplest
calculations.
Greek Geometry was codified by Euclid who expressed it in an axiomatic form
around 300 BC. But the axiomatic method was known earlier; it is mentioned by Aristotle
in his Metaphysics. Euclid’s original system was eventually found to be incomplete, and
its reliance on the use of diagrams left room for fallacious arguments, but the various
defects were repaired by David Hilbert working in the late nineteenth and early twentieth
centuries.
The axiomatic method was for many centuries treated as the paradigm of
systematic theoretical knowledge, and the hope that all knowledge might eventually be
cast in that form inspired Rationalism. Traditionally an axiomatic system was supposed to
start with a set of propositions whose truth could be easily established, they should if
possible be self evidently true. The remaining propositions of the discipline in question
were then to be deduced from the axioms.
Aristotle thought that the justification of the principles of Logic and the axioms of
Geometry was one of the functions of Metaphysics - the part of Philosophy presupposed
by all the individual disciplines, so an Aristotelian might offer that as an example of an
intrinsically philosophical problem, that can never be assigned to any more specialised
field of study.
Historical Summary: Mathematics and Logic
In the late 19th and early 20th centuries mathematically minded philosophers and
philosophically minded mathematicians extended logic and presented it in algebraic form.
That opened the way for a rigorous re-examination of the status of Mathematics. Even in
the eighteenth century Leibnitz had dreamt of a sort of arithmetic of logic, by means of
which the truth value of any proposition could be determined by some sort of calculation.
He hoped that we might get to a point where all disagreements could be settled in that
way. We’d now call this having a decision procedure. In propositional logic truth tables
provide just such a procedure. In modern terms Leibnitz thought that all propositions
might be decidable; that there might be some algorithm that could be applied to
absolutely any proposition and that could in every case be relied on to yield a definite
answer TRUE or FALSE.
Page 1
What is Philosophy Chapter 4
by Richard Thompson
At this point it is convenient to define algorithm. Any set of rules that can be relied
on to solve any problem of a certain type in a finite number of steps is called an
‘algorithm’. For example the standard procedures for addition, subtraction and
multiplication are all algorithms. In logical theory ‘decision procedure’ is equivalent to
‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew
or whatever it is that it tells us how to cook. A computer program, if it works, embodies
some sort of algorithm.
Only a Rationalist could have supposed there might be such a procedure for
determining the truth of any proposition, but even many Empiricists thought that the
propositions of Mathematics might all be decidable even if no other propositions were.
The Mathematician Peano (1848-1925) produced a short list of axioms from which
he hoped it would be possible to deduce all true propositions about the natural numbers,
the members of the set N={1,2,3...} I list those axioms later.
Frege tried to extend logic to the point where numbers could be defined within
logic, hoping to prove all Peano’s axioms and so to establish them as logical truths.
Russell found a contradiction in Frege’s system, but he and A. N. Whitehead published
Principia Mathematica (1910-1913 ) which described a system of logic that appeared to
be free from contradiction and sufficiently powerful to prove Peano’s axioms.
The ideal was a completely formal system in which inference started from certain
special propositions, the axioms, and proceeded by precisely defined rules, the rules of
inference. Once axioms and rules of inference have been chosen no subjective
judgement should be necessary. Purely mechanical tests should suffice to check whether
any proposed proof meets the criteria. A formalisation with that property is said to be
effective. Effectiveness is extremely important, for in everyday argument it is often hard
to be sure whether an argument is valid or not. As I have already remarked in chapter 2 it
can be especially hard to be sure that an argument is invalid, as that may require showing
that there is no valid form to which the argument conforms. Although we can be
completely sure an argument is invalid if it leads from a true premiss to a false
conclusion, that fact is no help in the cases where we do not know whether the
conclusion is true or false, and where we are interested in determining the validity of the
argument because, if valid, it will show that its conclusion is true. The desire to overcome
that difficulty was one of the motives behind the construction of formal logic.
Although the Russell-Whitehead system was formal in the sense that it had a
formal test for validity, it was not decidable - that is it did not have a decision procedure
that would determine, for an arbitrary formula, whether or not it was a theorem. It was
eventually proved that no system as strong as theirs could have such a decision
procedure. It appeared that the most one could hope for was that (1) Every truth
expressible in the system should have a proof and (2) There should be a decision
procedure capable of determining the validity of any putative proof.
However in 1931 Kurt Gödel (1906-1978) proved that even this is impossible. (1)
and (2) are incompatible so no consistent system strong enough to include even basic
arithmetic 1 can be both complete and effectively formalised. Since then a variety of
1
In this context ‘arithmetic’ does not mean just the carrying out of simple calculations, but includes the
whole of the theory of numbers, with algebraic identities and theorems about primes and divisibility.
Page 2
What is Philosophy Chapter 4
by Richard Thompson
formal systems have appeared sufficient to prove most of contemporary Mathematics, but
their construction is usually treated as part of Mathematics rather than pure logic.
I want now to describe the search for ‘foundations’ in more detail and consider what
wider significance the story may have. But before I do that I must introduce set theory and
equivalence relations.
Set Theory 
The beginnings of the subject are usually traced to Euler, who used diagrams to
illustrate inferences in Aristotelian logic. (see chapter 2 of these notes)
In 1847 George Boole (1815-64) published The Mathematical Analysis of Logic in
which he said “That which renders Logic possible, is the existence in our minds of general
notions, - our ability to conceive of a class and to designate its members by a common
name”. Boole developed an algebra of classes, using the arithmetical symbols for
addition and multiplication but specifying new rules so that ‘+’ represented union and ‘.’
represented intersection. I shall substitute for that the notation we use today.
Boole thought classes could be defined by specifying a common property peculiar to
the members of the class, and possessed by all members, thus {x: F(x)} denotes the
class of all individuals of which F(x) is true, so {x: x is a fish} is the class of all fish.
Although I can’t find a reference to the possibility in Boole, an alternative way of
defining a finite class is to list its members, as in U = {1, 2, 3, 4, 5, 6} In such a definition
the order in which the elements appear is unimportant. {4, 6, 1, 5, 2, 3} is just as good a
definition of U.
Membership: x is a member of set A is symbolised by ‘x  A’
The intersection of two classes is the class containing precisely the individuals that
belong to both classes, thus A  B= {x: x  A  x  B}.
for example {1, 2, 3, 4}  {2, 4, 6, 8} = {2, 4} The numbers are arranged in
numerical order for convenience, but listing them in a different order would not alter the
meaning.
The Union of two classes is the set containing just those elements that belong to
either of those classes or to both, thus A  B= {x: x  A  x  B} so that
{1, 2, 3, 4}  {2, 4, 6, 8} = {1, 2, 3, 4, 6, 8}
Notice that the numbers 2 and 4 appear only once in the listing of the members of
the union. Think of the union of two sets as like the merger of two clubs, where anyone
who used to belong to either of the original clubs becomes a member of the new club.
There might be people who originally belonged to both the original clubs, but such people
would not be members of the new club twice over with two votes at the AGM; they’d just
be members of it and have one vote each.
The Complement of a class. Boole refers to ‘the members of the universe’ that do
not belong to a class, A’ = {x: (x  A) } (x  A) is usually abbreviated to x  A)
Page 3
What is Philosophy Chapter 4
by Richard Thompson
To talk of ‘things that don’t belong to a class’ is obscure unless we say what sort of
things are eligible for membership. If G = the set of my garden tools, what is G’ ? Is it the
set of other people’s garden tools, or the set of all my tools that are not garden tools, or
the set of all physical objects that are not my garden tools? Are its members confined to
physical objects, or does it contain the numbers, and the plays of Shakespeare, and the
rules of etiquette of the Byzantine court in the tenth century? Until we’ve defined our
universe of discourse G’ is undefined. Today we usually represent the Universe class as
E and the empty class, that has no members, as  . The complement of a set S is then
interpreted as the members of the universe set that do not belong to S.
Equivalence relations
Equality has three important properties: it is:
(1) Reflexive a = a
(2) Symmetric a = b  b = a
(3) Transitive (a = b & b = c) a = c
any relation with those three properties is called an equivalence relation
If represents an equivalence relation over some set S,  is said to define a
partition of S. That means a subdivision into subsets S 1, ...Sn so that every member of S
belongs to one and only one subset and any two members, a and b, of the same subset
satisfy a  b . Those subsets are called equivalence classes.
In general a Partition of a set S is a subdivision of S into subsets with the properties
that:
(1) Every member of S belongs to precisely one subset, from which it follows that
(2) Any two sets in the partition are disjoint, that is they have no members in
common.
Any satisfactory filing system or classification system must be based on a partition
of the material to be classified, so that everything has its place, but only one place. To
any partition of a set there corresponds an equivalence relation over S, for the relation
that holds between two elements when they belong to the same subset is an equivalence
relation.
For example, define  as the relation holding between two natural numbers when
they both leave the same remainder when divided by 4. In the theory of numbers two
numbers so related are said to be congruent modulo 4.
Thus 5   
 partitions the natural numbers into four equivalence classes, one corresponding
to each of the possible remaindersand 3; the equivalence classes are :
{0, 4, 8,...}, {1, 5, 9,...}, {2, 6, 10,...} and {3, 7, 11, ...}
Another example of an equivalence relation is the relation of congruence over the
set P of plane geometrical figures. One of its equivalence classes is the set of all circles
of radius 15 cm. another is the set of all triangles with sides equal to 11 cm, 14 cm and
16 cm. There are infinitely many equivalence classes, one corresponding to each distinct
Page 4
What is Philosophy Chapter 4
by Richard Thompson
description of a plane geometrical figure.
Some theorists have defined numbers as equivalence classes of one sort or
another, and the most popular theories of the foundations of Mathematics base the
subject on set theory, so it seemed best to clarify both ideas at the outset, although
Peano, whose system I discuss first, didn’t use either sets or equivalence relations in his
treatment of natural numbers.
Peano’s System
Peano’s system, confusingly often referred to as Z, is sufficient to prove all the basic
rules of school algebra.
Peano developed the theory of natural numbers from their use in counting. For the
purposes of the present discussion, the natural numbers are the numbers {0,1,2,...} and
are called natural numbers because of their use in counting. Mathematical usage is not
completely consistent. For most purposes mathematicians apply the term ‘natural
number’ to the set {1,2,3...}, not including zero, but those who work in the foundations of
mathematics include zero in the set. During the first year of the operation of the Open
University I worked as a part-time tutor for the Foundation Course in Mathematics and
recall receiving several memoranda, alternatively saying that zero was a natural number,
and that it wasn’t. I don’t recall whether it finally ended up in or out of the set.
Peano took as primitive ideas the number zero, and the successor operation that
corresponds to the transition from one number to the next in the process of counting. The
successor of some number x is represented by x| . The numbers are represented in
Peano’s system by the symbols 0, 0|, 0||, 0|||, and so on. Symbols such as 1, 5, 73, are
treated as just abbreviations for a zero followed by the appropriate number of dashes.
Multiplication is represented by “.”
Z comprises:
Z1
(x)( y)(x| = y|  x = y) two numbers with the same successor are equal
Z2
(x)( 0  x|) zero is not the successor of any number
Z3
(x)( x + 0 = x) addition of zero makes no difference
Z4
(x)( y)( x + y| )= (x + y)|
Z5
(x)( x.0 = 0) multiplication by zero produces zero
Z6
(x)( y)( x .y| )= ((x.y) + x)
together with an infinite set of axioms for mathematical induction. The induction
rules are defined as all formulae of the form:
((P(0) & (x)( P(x) P(x|))) (x)(P(x))
that has the effect that if, for some quality P, it is possible to prove both that zero
has quality and P, and given also that, if P applies to any number, it also applies to the
successor of that number, then we may infer that P applies to every number.
Mathematical Induction Note that in Z mathematical induction is not a single
axiom, but an infinite set of axioms generated according to a rule. The principle of
mathematical deduction is that, as the natural numbers are defined by the process of
counting, a proof may follow the steps of that definition.
Page 5
What is Philosophy Chapter 4
by Richard Thompson
To prove (x)(P(x)) we first prove P(0), called the basis of the induction, and we
then prove that, if P(x) is true for any value of x, it must also be true for the number one
greater. That is called the induction step.
The two steps of the proof together establish that P is true of any number that can
be reached by starting from 0 and counting, in other words that P is true of any natural
number.
For example Induction may often be used to prove some formulae for sums of
series. The sum of the natural numbers from 0 to n is given by:
0 + 1 + 2 + 3+...+ n = [n(n+1)]/2
To prove that let 0 + 1 + 2 + 3+...+ n = Sn and let [n(n+1)]/2 = F(n)
The proof by induction is as follows:
Basis S0 = 0 and F(0) = [0*1]/2 = 0, so that S0 = F(0) .....(1)
Induction step suppose that for some number k, Sk = F(k) ..... (2)
(note that we’ve already shown that there is at least one such k, namely 0)
adding (k+1) to both sides of (2) gives:
Sk + (k + 1) = F(k) + (k+1)  Sk+1 = [k(k+1)]/2 + (k+1)
= (k + 1)[k/2 + 1]
= (k + 1)[(k+2)/2]
= [(k+1)(k + 2)]/2 which is the expression that
would be obtained by substituting (k + 1) for n in the original formula = F(k+1),
Hence the formula holds for every natural number n
Review of Peano’s Axioms
Z1 states that if two numbers have the same successor they are equal. Z2 states
that zero is not the successor of any number. It is possible to prove from the axioms that
every number except zero is the successor of some number.
Addition and multiplication are introduced by recursive definitions A recursive
definition of a function f is one that defines f(0), the value of the function for zero, and also
gives a rule for obtaining the value f(x|) from f(x). Z3 defines the addition of zero and Z4
specifies a rule that reduces the addition y| to the addition of y. The underlying idea is
illustrated by:
5 + 3 = 5 + 2| = (5 + 2)| = (5 + 1|)| = ((5+1)|)| = (5+1)||| = (5 + 0)|)|||
= (5+ 0)||| = 5||| = (0|||||)||| =...<a few steps omitted>...= 0|||||||| which is what we usually
abbreviate to “8”
Z5 defines multiplication by zero, and Z6 defines multiplication by y| in terms of
multiplication by y and addition.
Page 6
What is Philosophy Chapter 4
by Richard Thompson
A recursive definition lends itself particularly readily to proof by induction.
Note that Z2 precludes there being any natural numbers less than zero, so there are
no negative natural numbers. However that does not prevent the system being extended
to include additional objects to do the work of negative numbers, though such objects do
not satisfy the axioms for natural numbers.
One way of introducing negative numbers is to define integers in terms of ordered
pairs of numbers (a, b). Think of (a, b) as representing (a - b), so (5, 3) represents 2, and
(4, 8) represents -4. Thus many different pairs of natural numbers will represent what
we’d want to call the same integer, for instance (4, 8), (7, 11), (0, 4) all represent - 4
Define integer equality so that (a, b)  (c, d) if and only if a + d = b + c, then  is an
equivalence relation that partitions ordered pairs of natural numbers into equivalence
classes, one class corresponding to each integer.
Define integer addition by (a, b) PLUS (c, d) = ([a+c], [b + d])
Define integer multiplication by
(a, b) TIMES (c, d) = ([ac+bd], [ad + bc])
Just defining addition and multiplication like this is not sufficient to establish integer
arithmetic. We need to show also that the definitions are consistent and correspond to the
operations of addition and multiplication for integers.
We have defined integers as equivalence classes of pairs of natural numbers, so we
have to show that if we calculate, for example (-4) x(-3) we get the same answer
whichever number pairs we use to represent the -4 and the -3
In other words we have to show that sum and product as we have defined them are
not altered when either (a, b) or (c, d) is replaced by another equivalent number pair. I
shall do that in the case of addition.
We have to show that if (A, B) (a, b), and (C, D)  (c, d), then
(A, B) PLUS (C, D)  (a, b) PLUS (c, d)
(1) Suppose (A, B) (a, b), and (C, D)  (c, d),
(2) (A, B) (a, b)  A + b = B + a , and (C, D)  (c, d)  C + d = D + c
(3) (A, B) PLUS (C, D) = ([A + C], [B + D])
(4) (a, b) PLUS (c, d) = ([a + c], [b + d])
(5) from (1) and (2) A + b = B + a , and C + d = D + c
(6) from (5) A + b + C + d = B + a + D + c
(7) from (6) (A + C) + (b + d) = (B + D) + (a + c)
(8) from (7) ([A + C], [B + D])  ([a + c], [b + d])
(9) from (8) (A, B) PLUS (C, D)  (a, b) PLUS (c, d)
Hence PLUS defines an operation on integers.
That proof assumes that addition is associative and commutative. Those properties
are not explicitly stated in Peano’s axioms, but may be proved from them.
Page 7
What is Philosophy Chapter 4
by Richard Thompson
To show the connection between the integer operations just defined and the original
operations on natural numbers, we note that:
(a, 0) PLUS (b, 0)  (a+b, 0)
and (a, 0 )TIMES (b, 0)  (ab, 0)
so that every integer equal to an integer of the form (k, 0) corresponds to the
positive integer + k,
also (0, 0) corresponds to the natural number 0, since:
(0, 0) PLUS (a, b)  (a + 0, b + 0) = (a, b) and
(0, 0) TIMES (a, b)  (ax0 + bx0, bx0 + ax0) = (0, 0)
and (k, 0) PLUS (0, k) = (k, k)  (0,0)
so (0, k) corresponds to - k
We have now defined integers within Z, and can go on to prove that the product of
two negative numbers is positive, for
(-1)TIMES(-1) = (0,1)TIMES(0, 1) = ([0+1], [0 + 0])  (1, 0) = +1
Thus integers defined as pairs of natural numbers turn out to correspond to our
everyday notion of all the whole numbers, whether positive or negative.
There is also a number pair method for introducing rational numbers, the set of all
whole numbers and fractions, so a good deal of Mathematics can be proved from
Peano’s axioms, but, as Gödel proved, Z is nonetheless incomplete.
Logicism
It had long been generally believed that mathematical truths were a priori though
there was no general agreement what that amounted to. Empiricists thought
Mathematical truths must be analytic, but thinkers in the Platonic tradition disagreed. As
Mathematics has a close affinity to Logic, it occurred to some mathematicians and
logicians working in the late nineteenth century that Mathematics might actually be
reduced to logic. The thesis that it can be so reduced is called to ‘Logicism’.
Frege, although he didn’t use the concept of ‘equivalence relation’ suggested that
numbers could be identified with equivalence classes of sets. In 1884 he published Die
Grundlagen der Arithmetik in which he examined a number of popular accounts of
number. He was particularly concerned to rebut two common misconceptions.
(1) that a number is a property of physical objects, or collections of objects, and
(2) that arithmetical truths are inductive generalisations based on our experience.
Against (1) Frege pointed out that here is no unique number appropriate to any
object or collection of objects. One pack of playing cards is also 2 colours 4 suits, 13
denominations and 52 cards. Furthermore there is no need for objects to be physically
collected together to be numbered. If a couple have seven children, they have seven
children whether the children all live at home, or all in separate homes of their own.
A number, Frege argued, cannot belong to a physical object, or class of such
objects, but only to a concept. If there is just one pack of cards on the table the concept
‘Pack of cards on the table’ is associated with number 1, ‘Suit of cards on the table’ with
number 4 and ‘Playing card on the table’ with number 52.
Page 8
What is Philosophy Chapter 4
by Richard Thompson
Digression: Sets or Classes
In the late nineteenth and early twentieth centuries the terms ‘aggregate’ ‘class’
‘collection and ‘set’ were used interchangeably. Today ‘set’ and ‘class’ have been singled
out for use in special senses, but that was only after Russell had discovered an
underlying problem. In discussing the really stages in the attempt to develop a foundation
for Mathematics I shall follow Russell and Whitehead in using the work ’class’ .
Return to Frege and Russell
Frege made a proposal equivalent to specifying that (Natural) numbers be defined
as the equivalence classes of classes defined by the equivalence relation of one-one
correspondence. Two classes S and T were said to be of the same cardinality if their
members could be paired off so that every member of S is paired with precisely one
member of T and vice versa. (Frege attributed the idea to David Hume.) The natural
numbers were then identified with the equivalence classes defined by that relation. It
remained to produce a specimen class for each equivalence class so we can say which
natural number is which. Does that mean we need to assume the existence of objects of
some sort? three objects to define the number three and a million objects to define a
million?
Frege and Russell thought not. Even if nothing existed there would still be the empty
class, {} = , the class with no members.
Zero is the class of classes of the same cardinality as the empty class, . One is the
class of classes of the same cardinality as the class which has the empty class as its only
member, i.e. {}, two is the class containing zero and 1, namely {, {}}, and for any
natural number n, an example of an n membered class is the class containing all the
natural numbers less than n.
Thus an example of each natural number was to be constructed, ultimately, from the
empty class, with no need to assume that anything not a class actually existed. I suspect
that Frege and Russell thought that no justification was needed for assuming the
existence of the empty class, since it was established by definition, and that definition
does not appear to assume the existence of anything.
Russell’s Paradox
Frege’s original system was proved inconsistent by Russell, who derived a
contradiction from the supposition that there might be a class having as its members just
those classes that are not members of themselves. That paradox arises from the
seemingly innocuous assumption of extensionality, that for any predicate there is a class
of particulars of which that predicate is true.
Extensionality would imply that for any quality F there is a class { x: F(x)} the class
of objects that have quality F
{ x: F(x)} is called the extension of F. A predicate that does not apply to anything
has the empty class as its extension.
Russell proposed letting F(x) = x  x, ‘x is not a member of itself ’ so that the class
Page 9
What is Philosophy Chapter 4
by Richard Thompson
S = {x: F(x)} is the class of all classes that are not members of themselves.
Unfortunately to assert the existence of such a class is paradoxical, since if that
class S belongs to itself, it is not one of the classes that do not belong to themselves, and
so fails the test for membership of S, and so cannot belong to itself, while if S is not a
member of itself it satisfies the test and ought to be a member of itself.
Treating it formally, S = {x: F(x)} = { x: x  x)}
then S  S  F(S)  S  S giving a contradiction. S belongs to itself if, and only if,
it does not.
This shows that naive class theory, the collection of our common sense
expectations of classes (insofar as we have such expectations) is inconsistent.
In Principia Mathematica Whitehead and Russell listed seven paradoxes, remarking
that they were examples of an infinity of possible paradoxes.
Their first example, the paradox of the liar, is of great antiquity, originating with the
remark of Epimenides the Cretan that all Cretans were liars. It is usually presented in the
simpler form: P1: ‘This proposition is false’
If P1 is false, it follows that it is true, and if it is true it must, as it says, be false.
They also included Russell’s paradox of the class of all classes not members of
themselves, an analogous paradox about relations, the Burali-Forti contradiction about
the class of all ordinal numbers, which I discuss later in this chapter, and three paradoxes
about definitions of numbers of which I shall give just one example, known as Berry‘s
paradox.
Berry’s Paradox arises from the consideration of the definitions of numbers by
English sentences. In general larger numbers need longer definitions than smaller
numbers. That is only roughly true, because ‘seven hundred and forty three’ is a longer
phrase than ‘one million’, but as we consider progressively larger numbers even the
‘round’ numbers will need progressively longer definitions. Consider for instance
100000000000000000000000.
Let us define the number N so that
N = the smallest number that is too large to be defined in fewer than sixteen words.
Now consider the length of that definition; it contains only fifteen words which is less
than 16, contradicting the terms of the definition.
Whitehead and Russell (henceforth ‘W&R’) noted that the paradoxes had in
common the property of self reference and proposed to avoid such paradoxes by
introducing a Theory of Types, designed to prevent self reference.
W&R argued that the idea of something being a member of itself does not make
sense. Every individual should, they suggested, be assigned to a type. The basic type
contains individuals that may belong to classes, but cannot themselves have members.
Page 10
What is Philosophy Chapter 4
by Richard Thompson
Let’s call those individuals of type 0. A class of type 0 individuals would then be of type 1,
and so on. An object of type n may contain only objects of type n-1 and may belong only
to objects of type n+1. Thus no individual could sensibly be said to belong to itself. A
proposition of the form ‘S  S’ is just nonsense, and cannot be either true or false
because it doesn’t say anything.
I think the intuitive appeal of that argument is that we think of a class like a bag. A
bag may contain many things, even other bags, but it cannot contain itself, nor may a bag
B contain another bag that contains B, or contains anything that contains anything that in
turn contains B and so on.
Not all the paradoxes can be avoided by distinguishing types of individual. To avoid
the paradox of the liar, or of the number that can’t be defined in fewer than sixteen words,
Russell elaborated the theory of types by distinguishing different levels of language.
‘Grass is green’ would count as level 1, ‘It is true that grass is green’ would count as level
2 ‘It is not certain that it is true that grass is green’ would be of level 3, and so on.
So elaborated the Theory of Types did appear to avoid all the known paradoxes, but
only at the cost of considerable technical complications.
Although they said a great deal about classes, W&R thought that they were not
actually committed to the existence of such entities, insofar as the existence of a class
involves anything more than the existence of its members. The primary notion, they held,
was not class but the property used to define a class. References to classes were to be
given ‘contextual definitions’ in terms of the common properties of their members. The
only meaningful statements about classes were to be those covered by such definitions.
In isolation the symbol for a class had no meaning.
In accordance with the Theory of Types, only properties of a certain sort could be
used to define a class. They were extensional propositional functions.
A propositional function is what I have called an open sentence (see chapter 2),
something like ‘x is a fish’, which turns into a meaningful sentence when the name of
some physical object is substituted for the ‘x’. The class of fishes is then represented by
the formula {x: x is a fish}. (W&R actually used a different notation that is beyond the
capacity of my word processor.) However {x: x is a fish} has no meaning in isolation; only
when used in one of certain specified ways does it generate a proposition.
For example it can be used to generate the proposition:
{x: x is a fish} {x : x is a vertebrate}, which means the class of fishes is a subset of
the class of vertebrates.
The same proposition can be expressed as: ‘all fishes are vertebrates’ written as
(x)(x is a fish  x is a vertebrate)
Any finite class is easily defined in terms of identity. For instance x  {2,4,5} is
equivalent to (x = 2)  (x = 4)  (x = 5)
Page 11
What is Philosophy Chapter 4
by Richard Thompson
However not every propositional function defines a class. Basic to the idea of a
class is that it is completely specified when its members are specified, which is why W&R
required that the defining propositional function should be extensional. By that they meant
that truth should depend only what individuals are referred to. Thus
{x: the Bishop believes x is a good man} = the class of men whom the Bishop
believes to be good
does not define a class, because the Bishop might believe that the Vicar of Bray is a
good man, and also believe that the burglar who broke into the Episcopal palace last
night is not a good man, even though, unknown to the bishop, the burglar was the Vicar
of Bray.
‘The Bishop believes x is a good man’ is not extensional, because the truth of any
proposition generated from it depends not only on the identity of whatever individual is
substituted for the ‘x’, but on how that individual is described.
W&R found that their theory of types made it impossible to implement this analysis
of references to classes. To use classes as a basis for Mathematics they needed to be
able to refer to all the classes that have a particular individual, i, as a member. On their
analysis of classes, that involved referring to all the properties of any individual, but that
was forbidden by the theory of types which distinguished different orders of proposition
about any individual. All the theory of types allowed was reference to all the properties of
a particular order, so one could talk of ‘all the first order properties of i’, ‘all the second
order properties’, and so on, but not just ‘all the properties of i’
W&R defined the predicable properties of any individual as the properties of order
one higher than the individual. Thus green, hard, and light are all predicable properties of
physical objects, since all three properties are of the first order, while physical objects are
of order zero. On the other hand ‘colour predicate’ is a second order property since it
qualifies colours, which are themselves first order properties, so ‘colour predicate’ is a
predicative property of green and ‘green’ is a predicable property of grass, but ‘colour
predicate‘ is not predicable of grass. ‘Simon’s Shirt contains all the colours of the rainbow’
is not a predicable statement about Simon‘s shirt, since ‘colour of the rainbow’ is a
second order predicate, while ‘Simon’s Shirt’ is a basic individual of type zero.
W&R then introduced an Axiom of reducibility according to which, for every
extensional property of an individual, there is an extensionally equivalent predicable
property, in other words there is a predicable property that applies to precisely the same
individuals. Thus for any extensional property  there is a predicable property  that
applies in precisely the same cases so that (x)( (x)  !(x)) the exclamation mark is to
emphasize that  is predicable.
W&R’s example was ‘Napoleon had all the properties that make a great general”
That is a non predicable proposition about Napoleon as it refers directly to Napoleon’s
qualities not to Napoleon. The equivalent predicable proposition will attribute to Napoleon
a list of all the qualities that actually do make a great general, so it is something like
‘Napoleon was brave, cunning, cool, calculating, meticulous in his planning....’ I’m not
sure about that analysis. Napoleon’s possession of properties such as those must be the
justification for judging him to be a great general, but it does not follow that the judgement
is just a recitation of the considerations that justify it.
Page 12
What is Philosophy Chapter 4
by Richard Thompson
The apparent need to introduce the Axiom of Reducibility made it far less plausible
to claim that Mathematics had actually been constructed within logic, since the axiom had
been introduced specifically to facilitate that construction, not to meet any requirement of
logic in general. It may actually not even have been necessaryFor the axiom was
introduced to avoid problems created by distinguishing between different orders of
property, a distinction that had been made to avoid certain of the logical paradoxes.
However not all the paradoxes required that distinction. Those that did were the so called
semantic paradoxes, like the liar and the least number that can’t be defined in a certain
number of words. Those paradoxes arise from confusions about the meanings of words in
everyday discourse, and no such paradoxes could arise in a formal system like those
used in formal Logic and Mathematics. There is no way to translate into logical symbolism
‘This statement is false’ or ‘Number N cannot be defined in fewer than 16 words’ so there
is no need to complicate mathematical logic to avoid the semantic paradoxes, and
therefore no need to introduce the Axiom of Reducibility to ameliorate the complications.
Intrinsically logical paradoxes such as Russell’s Paradox can be avoided by the
simple theory of types. Quine later constructed a modified version of the W&R system
using only a simple theory of types. However most mathematicians interested in the
Foundations of Mathematics have rejected Logicism in favour of Formalism, a different
approach mainly due to David Hilbert (1862-1943). Formalists have usually preferred to
build Mathematics in set theory instead of in the predicate logic, so the next step is to
consider the development of mathematical set theory.
Zermelo Set theory
Working independently of Russell and Whitehead, their contemporary Ernst Zermelo
(1871-1953) tried to develop a set theory free from paradoxes by restricting the sets
assumed to exist.
Incidentally ‘set’ has come to be reserved for individuals that can both have
members, and can also themselves be members of other sets.
A collection that cannot itself be a member of anything else is called a ‘class’ or
often a ‘proper class’. There is no set of all sets, so ‘all sets’ is not a set but a proper
class.
Zermelo assumed some domain of objects with a relation so that a b asserts
that a is a member of b. His axioms were:
Zo 1: Two sets are equal if and only if they have the same members.
Zo 2: The existence of elementary sets.
2.1: There is a null set  with no members.
2.2: For every element x there is a set {x} of which x is the only member.
2.3: For any two elements x, y there is a set {x, y} containing x, y and no
other element.
Zo 3: Separation. If the predicate P is defined for all the members of a set S, then
there is a set consisting of all the members of S for which P is true; that set is
usually represented {x: P(x) & x S
Page 13
What is Philosophy Chapter 4
by Richard Thompson
Zo 4: Power set: For any set S there is a set consisting of all and only the
subsets of S
Zo 5: Union. For any set S there is another set, called the union of the members
of S, consisting of just those elements which belong to members of S
Zo 6: Axiom of choice. If S is a set of which all the members are non-empty sets,
then the union set of S has at least one subset which has exactly one member in
common with each element of S
Zo 7: Axiom of Infinity: There is at least one set, Z, such that one of its members
is the null set, and if x belongs to Z, so does {x}
How far are these axioms intuitive in the sense of expressing our concept of set?.
The difficulty is that until we study Logic or Mathematics we don’t have much intuition
about sets, and any sense of intuition we feel later may just be a reflection of the climate
of mathematical opinion at the time we learnt our set theory. I think people usually
imagine a set either as a collection of objects, or as a list of objects. Both pictures are
liable to be misleading. ‘Collection’ is misleading because there is no need to do any
physical collecting, and not even any possibility of doing it in the case of abstract
objects. ‘List’ doesn’t apply to infinite sets. Georg Cantor (1845-1918) said “A set [he
used the word Menge ] is a collection into a whole of definite distinct objects of our
intuition of our thought” Nonetheless I think that when people find at least some of Zo1 to
Zo7 intuitively plausible, it is because they are thinking of either collections or lists. As I’ve
already remarked, I suspect that what Mathematicians and Logicians call their ‘intuitions’
about sets are really their intuitions about bags, and that there are really no intuitions
specifically about sets.
An alternative is to think of set theory as a game invented by Zermelo and others,
and to say ‘ Can a game be played according to these rules; are they consistent?’ For
any formal system consistency can be established by producing a model. (discussed
later) Finite sets can be modelled by either collections or lists so for Zo1 to Z05 we can
safely rely on intuition even if it is derived from those defective pictures of set theory.
I’m quite unable to conjure up much of an intuition for Zo7. Axiom 7, the axiom of
infinity corresponds to Frege’s construction of specimens of the numbers as , {}, {,{}},
{,{}, {{}}}, ... so we definitely need it, but what intuitive appeal it has seems to me no
more than our desire to get the natural numbers.
Zo3, which permits the definition of sets of elements possessing a common quality,
does not permit the definition of the set of all sets not members of themselves, because
the axiom permits only the definition of a new set that is a subset of some set already
established, thus Zermelo’s system avoids Russell’s paradox.
Zo6 is controversial, and Mathematicians like to go as far as they can without using
it, but it is at least obviously true in the case of finite sets.
Although they are not explicitly mentioned in any of the axioms, the three simple
binary operations on sets, union, intersection and difference are provided for.
Page 14
What is Philosophy Chapter 4
by Richard Thompson
Given any two sets S and T,
Union: Zo 2.3 guarantees a set {S, T} and Zo5 entails that there is a set R
containing precisely the objects that belong to either S or T. R is ST
Intersection: In Zo3 let P(x) be ‘x  T’ then
{x: P(x) & x S = {x: x  T & x S = S  T
Difference: In Zo3 let P(x) be ‘x  T’ then
{x: P(x) & x S = {x: x  T & x S = S - T = S  T’
The existence of a set {x} for any x, together with the existence of a union for any
two sets, guarantees the existence of a set containing any finite list of elements we care
to specify and also, therefore, the existence of a union for any finite list of sets. However
the union of an infinite list of sets is not available unless those sets are themselves
members of some set, and there is no guarantee that they will be. For instance there
cannot be a union of all sets, because there is no set of all sets. The Axiom of
replacement (see below) allows unions for some infinite lists of sets.
Zermelo’s system has since been tidied up in various ways, mainly by Fraenkel and
the amended system is usually referred to as ‘ZF’
Von Neumann later proposed an axiom of Foundation, to rule out infinitely
descending chains of sets like: x1 x, x2  x1,...xn+1 xn....
The axiom of foundation states that every non-empty Set S must contain an element
T, such that S and T have no common element. That prohibits the identification of a one
membered set with its only member, since if x = {x} we should have x  x and x would
have no member disjoint from x. That consequence was unacceptable to Quine whose
liked to avoid postulating abstract entities unless it was absolutely necessary.
If we do identify a one membered set with its only member, we can construct an
infinite descending chain by setting xn = x for all n. In the light of that simple example the
idea of an infinite descending chain no longer seems counter intuitive. When I first
encountered the idea of an infinite descending chain I found it baffling because each set
in the chain appeared to have no definite membership, because at least one of its
members was another set with the same frustrating characteristic. However that feeling
arose because I was treating the description of the chain as a definition of its members.
Were the members of the chain defined in some other way, from which their relative
positions in the chain followed, there need be nothing puzzling about the relationship.
I have not been able to discover enough of the background to the subject to find
why infinite descending chains were considered objectionable. Was it that some proof
emerged of the existence of an unwelcome chain? Or was it just that someone realised
that a chain was a theoretical possibility not inconsistent with the basic Zermelo axioms,
and recoiled from mere the possibility ?
The fact that the axiom had to be added as an afterthought underlines the
weakness of our intuitions of sets. Furthermore if infinite downward descent is
undesirable, that seems to weaken the case for the (essential to Mathematics) infinite
Page 15
What is Philosophy Chapter 4
by Richard Thompson
ascent of Zo 7.
Another addition, due to Fraenkel, was the Axiom of Replacement, that if S is a set,
then replacing the members of S by anything in the domain produces another set. So that
is S is a set, and R is a function with a unique value for every member of S, there is also
a set T = {t: t = R(s), s  S}
The Axiom of Replacement was introduced to allow the definition of more sets. Zo 7
guarantees one infinite set, Z, and replacement applied to Z gives more. In particular the
union of all the sets in an infinite list is now available. Fraenkel was particularly interested
in allowing the unions of certain infinite sets, the members of which were themselves
infinite sets, and some distinctly counter-intuitive consequences concerning large infinite
cardinals eventually emerged.
While Peano’s axioms could be said to follow from our idea of number because they
agree with our intuitions, the same cannot be said for all the axioms of Zermelo set theory
and some of the proposed extensions to that theory seem even less intuitive.
Formalism
While Frege, Whitehead and Russell were trying to construct Mathematics within
Logic, Hilbert worked to reduce Mathematics to a collection of formal systems, in which
mathematical Logic was augmented by specifically mathematical axioms.
The background to Formalism was axiomatic geometry, which was the starting point
of Hilbert’s own work.
For two millennia Euclid’s axiomatization of geometry was accepted without
challenge, but doubts arose during the nineteenth century. Mathematicians were
particularly concerned about the parallel postulate which guarantees that, given a straight
line L and a point P not in L, there is a unique line through P parallel to L.
Attempts to prove the parallel postulate from the other axioms failed, and it was
eventually shown that it could not be proved since replacing the postulate by other
axioms did not give a contradiction but instead produced alternative geometries, in which
there were either no parallel lines at all, or more than one line through P parallel to L.
Some of those geometries were locally indistinguishable from the Euclidean.
A familiar example of a non-Euclidean geometry is spherical geometry, treating of
figures on the surface of a sphere. In spherical geometry the place of straight line is taken
by the great circles, namely the circles with centres at the centre of the sphere. On a
globe the lines of longitude are great circles through the poles, and the equator is also a
great circle. The shortest distance between two points, measured in the surface of the
sphere, lies along the great circle joining the points, so aeroplanes often fly along great
circle routes. There is usually precisely one great circle joining two points in the surface of
a sphere, except when the two points are at the opposite ends of a diameter, like the
North and South poles, where there are infinitely many great circles joining them. The
great circles are the nearest equivalent in spherical geometry to the straight lines of
Euclidean Geometry. However, any two different great circles meet in two points, so there
are no parallel lines in spherical geometry, and in a spherical triangle, formed by the arcs
of three great circles, the sum of the angles depends on the area, and is in all cases
Page 16
What is Philosophy Chapter 4
by Richard Thompson
greater than the 180 degrees which is the sum implied by the Euclidean axioms.
o
If N is the North Pole, and A and B are the two points on the equator, longitude 0 at
o
A and longitude 90 W at B, the spherical triangle NAB has all three of its angles right
o
angles so that its angle sum is 270 . Yet if we confine ourselves to a relatively small part
of the earth’s surface, such as a town, we can treat the surface as approximately a plane
and apply Euclidean geometry. If we mark out a triangle on the school playing fields and
measure its angles we should be unlikely to find their sum significantly different from
o
180 ; that does not imply that the earth is flat, just that the playing fields are small enough
for plane geometry to be an adequate approximation. Similarly the fact that Euclidean
geometry seems adequate for everyday purposes does not imply that it is universally
true, just that it is adequate to describe the part of space we inhabit sufficiently accurately
for our purposes.
In principle the non-Euclidean geometries could be tested by measuring the three
angles of a triangle. The various non-Euclidean geometries imply sums differing from 180
degrees by quantities that increase with the area of the triangle. No appreciable deviation
from 180 has been observed in the conventional geometry of space, but every physical
measurement is subject to some error and the best we can do is to show that the angle
sum of every triangle we have actually been able to measure differs from two right angles
by less than the margin of error in the measuring procedure. So if we regard geometry as
describing the space we live in, we can’t be sure that the parallel postulate is true, and
even though we are not sure of that we can still describe our experiences perfectly. That
refutes Kant’s claim that Euclidean geometry is a body of synthetic a priori truths, so
called because they are true in every conceivable world, for we can conceive worlds in
which Euclidean geometry would be false, even though we cannot be sure that ours is
such a world.
There seems to be a choice between regarding Mathematics as a body of
contingent truths describing the world, or as a body of necessary truths describing
abstract entities analogous to Plato’s Forms. Logicism provided one way of avoiding both
those alternatives, but not the only way. Hilbert suggested another way by proposing that
Mathematics might be developed as a study of formal systems without actually specifying
any meaning for them.
Formal Systems and Interpretations
One could think of a formal system as a game played with symbols according to
rules allowing us to progress from some combinations of symbols to others in certain
ways. Such progressions correspond to mathematical proofs.
It is possible to interpret such a game by setting up a correspondence between
formulae and their transformation rules on the one hand, and some set of individuals and
the rules for manipulating them on the other.
As an example take the propositional logic. In chapter 2 I introduced the logical
symbols by saying what each symbol meant, but it would be possible just to tell someone
the rules for constructing truth tables or proofs, without giving any interpretation. That
would be like programming a computer to produce truth tables and to say ‘tautology’ or
‘contradiction’ where appropriate. We should then have an uninterrupted formal
Page 17
What is Philosophy Chapter 4
by Richard Thompson
system.
When we construct a formal system it is because we intend to use it to represent
some body of knowledge, and we therefore have in mind particular meanings for the
various symbols involved, but formalists thought it essential that the rules for operating
the system make no appeal to those meanings or to the intended application of the
system. Several quite different interpretations of the same system are usually possible;
sometimes there are even several different interpretations that we find useful.
For instance in the case of the propositional logic, the interpretation of the logical
symbols that logicians had in mind when they originally invented the system is only one
interpretation among many. A purely arithmetical interpretation is possible.
Let P, Q, R...be numbers taking either of the values {0, 1}
Let P = 1- P, P & Q = PQ, (P V Q) = P + Q - PQ, (P  Q) = 1 + PQ - P, and
(P  Q) = PQ + (1 - P)(1 - Q).
The tautologies are then the formulae that equal 1 for all possible combinations of
values of their variables, and the contradictions are the formulae equal to 0 for all values.
It is also possible to interpret the truth functional logic as an algebra of sets, with &
representing intersection and V representing union.
The axioms of geometry also admit of alternative interpretations. By inventing
co-ordinate Geometry Déscartes showed that they can be interpreted arithmetically.
The Properties of a Formal System
A formal system usually consists of:
(1) A set of symbols
(2) Syntactic (grammatical) rules specifying which formulae are properly constructed
- such formulae are called well formed formulae (wff’s for short)
(3) A set of wff’s called the axioms. If the set of axioms is finite they may be listed,
but if they are infinitely many there must be a decision procedure (algorithm) that will
determine, for any arbitrary wff , whether or not it is an axiom
(4) A set of Rules of Inference that allow one to construct theorems. All axioms
are theorems, and the rules of inference allow new theorems to be derived from ones
already known.
The propositional calculus is a particularly simple formal system since there is a test
(constructing the truth table) to find out which formulae are theorems. Such a test is
called a decision procedure.
If a formal system has a such decision procedure, it is possible to dispense with (4)
by making all the valid formulae into axioms. That is what we were doing when we
defined tautologies in terms of truth tables. Whitehead and Russell originally gave a set of
five axioms and various rules of inference from which all the tautologies could be
Page 18
What is Philosophy Chapter 4
by Richard Thompson
deduced.
Consistency
We always require systems to be free from contradiction. One way of showing that a
system is consistent is to produce a model (interpretation) of the system to show that it
can be used to represent something that is clearly consistent. I shall give a detailed
specification of a model of a formal system later, but for the moment a simple example
should suffice to show what is involved. The arithmetical interpretation of the propositional
logic in terms of calculations with the numbers 1 and zero showed that the propositional
logic was consistent. To demonstrate consistency like that the model has to be simple
enough for us to see that it is consistent, which is easiest if the interpretation is finite.
However Peano’s system and all the stronger systems developed to accommodate
progressively more mathematics, have no finite models, they were after all constructed to
guarantee the existence of infinitely many numbers.
However, in the absence of a simple interpretation there is another way to establish
consistency.
In a system that includes basic logic, a contradiction implies anything at all, so that
in a contradictory system every wff will be a theorem. So to show consistency, it suffices
to find one wff that is not a theorem. The strategy is to find some property satisfied by all
the axioms, and preserved by all the rules of inference, and then to find some wff that
does not possess that property. Often a convenient property is truth under some
interpretation but any property would do.
One might suppose that to prove the consistency of a formal system S should
require some even stronger system S*, so that if the consistency of S is doubtful, that of
S* must be even more dubious, making the proof of little value. However a consistency
proof need consider only a few properties of a formal system, so even though S* may be
stronger than S in the single respect of being able to prove the consistency of S; it might
in other respects be much weaker, or so the Formalists hoped.
In the early twentieth century it was hoped that consistency proofs might be finitary
in the sense “that the discussion, assertion, or definition in question is kept within the
bounds of thorough-going producibility of objects and thorough-going constructibility of
processes, and may accordingly be carried out within the domain of concrete inspection”
(Hilbert and Bernays Grundlagen der Mathematik) Hilbert and members of his school
looked for proofs such that (1) they were constructive in the sense that any entities
referred to could be produced for our inspection (2) all processes involved were
guaranteed to be completed in a finite number of steps, that number always being within
some bound that is known in advance.
Initially the formalists made encouraging progress, proving the consistency of the
propositional logic and the first order predicate logic.
In the case of the propositional logic we need consider only two truth values for
n
each variable, and the truth table for any formula contains precisely 2 rows. That was
how I defined truth functions, but some systems introduce truth functional logic
axiomatically. Such systems can be shown consistent by showing first that every axiom
has a truth table with a ‘T’ in every row, and second that the rules of inference preserve
Page 19
What is Philosophy Chapter 4
by Richard Thompson
that property, so that anything deduced from a tautology is itself a tautology.
A similar procedure can be used for a subset of the predicate logic. Consider
formulae involving just one place predicates. Consider for example a formula that
contains two such predicates, as in

x )(F(x)) & (x)( G(x) )]  x )(F(x) & G(x))
I shall show this subset of predicate logic is consistent by showing that that formula
is not a theorem.
In assessing the truth of the various components of that formula it suffices to
consider individuals of just four different kinds. Those that are both F and G, call them
FG’s, those that are F but not G, call them Fg’s, those that are not F but are G, fG’s, and
those that are neither F nor G, fg’s.
If there are any FG individuals, every component proposition is true so (1) reduces
to T  T, which is true.
If all the objects in the domain are fg, every component proposition is false, so (1)
reduces to F  F which is again true.
However, if the domain contains no FG’s, but does contain both Fg’s and fG’s
x)(F(x)) and (x)( G(x) ) are both true, but x )(F(x) & G(x)) is false, so (1) reduces to
T  F which is false, so (1) is not a theorem.
On the other hand
(2) x)(F(x))&(x)(G(x))]  x)(F(x)&G(x)) is valid.
It could only be false if x )(F(x)&G(x)) were false while x)(F(x))&(x)(G(x)) were
true,
but ifx )(F(x) & G(x)) were false there must be at least one object that is not FG,
and hence either Fg, or fG or fg,
in which case at least one of x)(F(x)) or (x)(G(x) must be false, making
x)(F(x))&(x)(G(x)) false so that the whole expression would be true.
In that example I abbreviated the procedure by grouping together interpretations
that should strictly speaking have been considered separately. As there were four
different types of object (FG, fG, Fg, and fg), a full treatment would have required
consideration of fifteen different possibilities, since each of the four types could be either
4
present or not present in the domain, giving 2 significantly different types of domain of
individuals, from which we must subtract one to exclude the possibility that none of the
four types is present at all, which would imply an empty domain.
With a decision procedure for theoremhood, it is easy to prove consistency - we
have already done so for the one place predicate logic by showing that formula (1) is not
Page 20
What is Philosophy Chapter 4
by Richard Thompson
a theorem.
No such decision procedure is available for logic with two place or more place
predicates, for once we introduce a predicate such as L(x,y) = x loves y there are infinitely
many different types of individual. There are people who love no-one, people who love
one person, people who love two people, and so on, giving an infinity of types even
without considering more complicated cases like people who love three people, two of
whom reciprocate their feelings, and are loved by seven other people none of whose
affections they return.
However we can show the consistency of the predicate logic, even with multi-place
predicates as follows. Simplify all formulae by omitting all quantifiers and variables, and
treat all the predicate letters as if they were propositional variables. It is possible to show
that when so treated, every valid formula is turned into a tautology. For instance
(x)(P(x)) reduces to P, which is not a tautology, so (x)(P(x)) is not a valid formula
The tautologies so obtained are not, of course, equivalent to the original formulae.
Some formulae that are not valid will reduce to tautologies so this procedure is not a
method of establishing that individual formulae are valid. For instance the formula:
(1) x )(F(x)) & (x)( G(x) )]  x )(F(x) & G(x))
which I discussed above yields a tautology, even though it is not valid. The
important thing is that every valid formula will be reduced to some tautology. Therefore
any formula that does not reduce to any tautology is not valid. As there are some
formulae that don’t reduce to tautologies, there are some formula that can thus be shown
not to be provable, so the system is consistent. In particular there cannot be two provable
formulae related in the same way as F and F since no two formulae so related could
both reduce to tautologies. If one reduced to a tautology T, the other would reduce to T,
which would be a contradiction.
Proving the consistency of arithmetic turned out to be much harder. Consistency
proofs of considerable complexity were produced for some fragments, notably for the
arithmetic of natural numbers using addition but not multiplication, but the enterprise
came to a halt when Gödel proved that any system that can prove its own consistency
must in fact be inconsistent. To be more precise he proved that if, in a system S, it is
possible to construct a formula F, such that F is true if and only if S is consistent, then F is
provable in S if and only if S is inconsistent. He also showed how to construct such a
formula F for arithmetic.
At first glance that might not appear particularly worrying. Consistency proofs
always had been performed outside the system being proved consistent. In any
inconsistent theory all formulae are provable, so if an inconsistent theory can express its
own consistency it will also be able to prove it. So a consistency proof constructed within
a theory would not provide any reason for believing that the system actually is consistent.
The discovery that many systems can only prove their own consistency if they are
inconsistent was both surprising and interesting, but even without that discovery it should
have been plain that, if the consistency of a theory was in any doubt, a consistency proof
in that same theory would have been worthless.
Page 21
What is Philosophy Chapter 4
by Richard Thompson
Oddly enough Formalists had hoped to construct consistency proofs using only very
simple inferences which would be a subset of arithmetic. Hence the proofs the formalists
sought would in the case of arithmetic have been capable of incorporation in the system
being proved consistent and therefore of the type that should have been perceived to be
useless even before Gödel rammed the point home.
Thus theories such as Peano’s and stronger theories can only be proved consistent
in theories in all respects stronger than themselves, which makes the formalist program
seem no longer very interesting.
Algorithms and Computers
A formal system is said to be effectively axiomatisable if there is a decision
procedure for deciding whether or not an arbitrary formula is an axiom, and a decision
procedure for deciding whether or not any proposed inference conforms to the rules of
the system. A satisfactory decision procedure must be one that can be relied upon to give
a definite answer in every case after only a finite sequence of steps. Such a procedure is
an algorithm.
Church’s Thesis (named after Alonzo Church (1903-1995)) says that the set of
algorithms is identical with the set of problems that can be solved by computer. No one
has been able to offer a proof for Church’s Thesis, it’s hard to know where one would
start, but the thesis is thought to be highly plausible, especially as it has been shown that
the set of computable functions is the same as the set of recursive functions. Logicians
usually discuss computing in terms of an idealised rather simple computer called a
Turing Machine, named after Alan Turing. I’m not sure whether anyone has ever actually
made a Turing machine, but such machines can be emulated on the much more
complicated computers that we actually do use. It has been shown that any recursive
function can be evaluated on some Turing machine
Gödel and Incompleteness
During the early decades of the 20th century Mathematicians still hoped that the
whole of Mathematics might eventually be represented by a formal system until, in 1931,
Kurt Gödel proved that this was impossible.
He showed that any effectively axiomatisable system capable of expressing even a
substantial part of Mathematics, must either be inconsistent, or else incomplete, in the
sense that its notation must permit the expression of a true proposition that cannot be
proved in the system. Inconsistency is of course completely unacceptable, since, as
we’ve already seen, a contradiction entails anything whatever, so that in an inconsistent
system every wff would be a theorem and there would be no distinction between truth and
falsehood. We therefore have to accept that any formal system for Mathematics will be
incomplete.
Gödel’s strategy was to construct formulae that had a sort of double meaning.
Within the formal system they were mathematical statements, asserting that a certain
equation has no solutions, but considered from outside the system they could be seen as
asserting their own unprovablity.
Gödel’s first step was to define what has come to be called a Gödel numbering of
the formal system in which every symbol is assigned a number, and there is a rule such
that, given any sequence of symbols, a number can be calculated for the sequence from
Page 22
What is Philosophy Chapter 4
by Richard Thompson
the numbers of its components. No two sequences of symbols have the same Gödel
number, so any Gödel number can be decoded to give the original sequence.
Gödel then defined a proof predicate P(x,y) to represent provability in the following
way. If A is a series of formulae that constitute a proof of some formula B, and if the
Gödel numbers of A and B are g(A) and g(B), respectively, then
P[g(A), g(B)] is true if and only if A is a proof of B
‘Formula B is provable’ can then be presented by (x)[P(x, g(B)] and
‘Formula B is not provable’ by (x)(P(x, g(B))
Define ‘Formula(x)’ = the formula with Gödel number x
Define the diagonalisation of a formula A as: (x)[A=Formula(x) & A]
Suppose now that ‘A’ contains ‘x’ as its only free variable, the diagonalisation of A
then asserts that A applies to its own Gödel number.
Gödel showed that there is a recursive function n  diag(n) that calculates the Gödel
number of the diagonalisation of the formula A that has n as its Gödel number.
Now consider (k)(P(k, diag(x)) which asserts that diag(x) is not the Gödel number of a
provable formula
Let M = g((k)(P(k, diag(x)), so that M is the Gödel number of (k)(P(k, diag(x))
So Formula(M) = (k)(P(k, diag(x)), and Diag(M) = (k)(P(k, diag(M))
so that Diag(M) is the Gödel number of a true formula if and only if Diag(M) is not
the Gödel number of a provable formula. M is provable only if it is false, so that it is
provable only in an inconsistent system. Yet if it is not provable, it is true. So any
consistent system powerful enough to define Diag must be incomplete.
An important consequence of Gödel’s incompleteness theorem is that we cannot
define Mathematics by giving a formal system, since no effectively axiomatisable system
can contain the whole of Mathematics.
Gödel’s incompleteness theorem applies to any effectively axiomatisable system at
least as strong as a certain system Q, where Q is defined as the predicate logic combined
with the following mathematical axioms:
Q1
(x)( y)(x’ = y’  x = y)
Q2
(x)( 0  x’)
Q3
(x)( x  0 ( y)(x = y’))
Q4
(x)( x + 0 = x)
Q5
(x)( y)( x + y’ = (x + y)’)
Q6
(x)( x’.0 = 0)
Q7
(x)( y)( x .y’ = (x.y) + x)
The notation is the same as that used in Peano’s axioms. x’ means the successor of x. ‘
Page 23
What is Philosophy Chapter 4
by Richard Thompson
is the successor operation that generates any number except zero from its predecessor.
Q is weaker than Peano’s system as it does not include mathematical induction, but it
does include all the other axioms of Z and also includes Q3 which is a theorem of Z, but
which, in the absence of the induction axioms, cannot be proved from the other axioms in
Q.
All the individual arithmetical facts are provable in Q, but the commutative rules for
addition and multiplication are not, so Q is a relatively weak system, but it is strong
enough to define all the recursive functions and hence powerful enough to support the
definitions of the functions Gödel used in his proof.
Gödel’s theorem has no direct bearing on the parts of mathematics with which most
people are familiar. Simple arithmetical truths, of the form a + b = c, and a x b = d can all
be proved, they all follow from Peano’s axioms, and indeed from the weaker system Q.
All the elementary algebraic formulae follow from Peano’s axioms. It is only algebraic
propositions about unspecified numbers that are sometimes undecidable, and of those
only propositions involving both addition and multiplication. There is a complete system
for numbers with addition but no multiplication, and another complete system for numbers
with multiplication but no addition.
The Gödel sentences are in fact all true. The undecidable sentences for which
Gödel provided a method of construction assert that a certain equation has no solutions.
The equation is an eighth degree polynomial in eighty variables, with at least one of its
coefficients some thousands of (decimal) digits long. Different formal systems will give
rise to different equations, but all are of the form just specified.
If such an equation actually has a solution, its solubility is provable in Q, just by
exhibiting the solution and verifying by calculation that it is correct. To do that requires
only the ability to perform arithmetical calculations for which even system Q suffices.
Hence the proposition that an equation has no solution could be disproved if it were false,
so, if undecidable, it must be true.
Of course Gödel’s result does not show that there are mathematical truths that
cannot be proved at all. What it shows is that, given any system S for arithmetic, there is
some truth that cannot be proved in S. It is important to note that Gödel’s proof is
constructive, that is it shows how, working in a metalanguage, one can construct for any
effectively axiomatisable formal system an undecidable but true proposition. Once we
have the undecidable proposition, call it U, we can add it to our formal system as an
additional axiom, giving a new stronger system in which U is (trivially) a theorem. In that
stronger system U itself would no longer be equivalent to the assertion of its own
unprovablity, but there would instead be some other proposition that would be
undecidable in the stronger system in the same way that U was undecidable in the
weaker.
Since Gödel proved the theorem that bears his name, other mathematicians have
produced more undecidable sentences that, while still rather abstruse, are less remote
from ordinary mathematical practise than the original Gödel sentences. There are many
references to such sentences on the World Wide Web, but I have been unable to obtain
sufficient details to give examples here.
Mathematical Truth
Page 24
What is Philosophy Chapter 4
by Richard Thompson
Since no effectively axiomatisable formal system can be complete, Mathematical
truth cannot be identified with provability in any particular formal system, so we need an
alternative definition.
The alternative is truth under any standard interpretation.
To explain that we must start with interpretation. The use of the word is not entirely
consistent. There is a loose sense in which it means giving some meaning to the symbols
of a system. The looser sense is applicable when logicians demonstrate the mutual
independence of axioms by constructing interpretations in which all but one of the axioms
true and the remaining axiom is false. In the present context ‘interpretation’ is used more
strictly to mean giving the symbols an interpretation that make all the theorems true.
In that stronger sense an Interpretation of a formal system is:
(1) A domain of individuals
(2) an assignment of one of the individuals to every individual constant, or, as it is
sometimes put, the assignment of a designation for every name
(3) an assignment of a truth value (TRUE or FALSE) to every propositional constant
(4) an assignment to each functional letter of a function mapping (some) members
of the domain to members of the domain.
(5) a characteristic function for each function letter chosen so that it provides an
assignment of truth conditions to every predicate letter in such a way that:
(a) For each one place predicate letter F, the interpretation specifies for which
individuals F(x) is true
(b) For each two place predicate letter G, the interpretation specifies for which
ordered pairs of individuals G(x, y) is true
(c) and so on for three or more place predicate letters.
Thus an interpretation of (x)[F(x)  G(x, x)]  [F(a)  G(a, a)]
must assign some particular individual to a, and must provide algorithms for determining,
for every individual x, whether F(x) is true or false, and for every pair of individuals, x, y
whether G(x, y) is true or false.
A theory is a set of sentences that includes the logical consequences of every
member of the set - in other words a set of sentences closed under logical consequence.
An interpretation satisfies a sentence if that sentence is true under the
interpretation.
A model of a sentence is an interpretation that makes the sentence true. A model of
a theory makes every axiom and theorem true
P entails Q if Q is true under every interpretation that makes P true.
A formula is valid if it is true under every interpretation. A valid formula will
represent a logically true proposition.
We now have a definition of logical truth, but the definition does not always suffice to
determine whether a particular sentence is valid or not, though it is sometimes possible to
settle the logical status of a sentence from first principles by considering possible
Page 25
What is Philosophy Chapter 4
by Richard Thompson
interpretations. To be valid a sentence must be true under all interpretations, so we can
show that a sentence is not valid by finding just one interpretation that makes it false.

For example (x)(F(x))  (x)(F(x)) can be shown invalid using an interpretation
with just two elements - call them a and b
Let F(a) be true and F(b) be false
Then (x)(F(x)) is true but (x)(F(x)) is false so that (1) reduces to T  F, which is
false.
However, while falsehood under just one interpretation establishes invalidity, to
prove that a formula is valid requires that we show that it is true for every interpretation.
Since there are infinitely many possible interpretations we can do that only in a few
special cases. In simple cases it may be possible to group the infinitely many
interpretations into finitely many classes, so that all the interpretations in the same class
assign the same truth values to every sentence. As we’ve already seen that can be done
for the predicate logic restricted to one place predicates where, if there are n predicates,
n
the number of significantly different interpretations is 2 - 1
The Standard Interpretation
That is the interpretation in which the symbols are interpreted in their ordinary
mathematical senses, with ‘0’ representing zero, 0| representing 1, + representing
addition and . representing multiplication. Mathematical truth is then truth for every
standard interpretation.
Non-Standard Interpretations
Because any effectively axiomatised system that contains Q is incomplete, there will
be mathematical truths that cannot be proved in it. It therefore has models in which the
sentences corresponding to those unprovable truths are false. Such a model is called a
non-standard model. Non-standard models contain extra elements in addition to the
natural numbers.
Even in a system as weak as Q, the axioms guarantee the presence of 0, 0’, 0’’,....
so something like the natural numbers will be present in any model of the axiom set, but
there is nothing to prevent other elements appearing too.
Frege’s Ancestral relation
Frege held that it was important to construct a definition of number that did not apply
to anything except the natural numbers, and proposed to eliminate non-numbers by
specifying that nothing is a natural number unless it can be reached in a finite number of
steps, starting from zero. That is easily said in ordinary language, but not so easily
formalised.
Frege developed what is generally referred to as ‘Frege’s Ancestral Relation’
Suppose we want to analyse ‘x is an ancestor of y’ in terms of ‘x is a parent of y’ ‘x
is an ancestor of y’ means: ‘y has all those properties that apply to x and are always
transmitted to their children by anyone who has children’
Analogously we could in arithmetic define ‘n is a descendant of zero’ as
‘n has all the properties that apply to zero and are such that, if they apply to x, also
Page 26
What is Philosophy Chapter 4
by Richard Thompson
apply to the successor of x’
i.e. (P)([(P(0) & (x)(P(x)  P(x|)] P(n)) ....(1) where P represents a property
‘n is a natural number can then be written:
(n = 0) V (P)([(P(0) & (x)(P(x)  P(x|)] P(n))
‘n is zero or n has all the properties of zero that, if possessed by any number, are
also possessed by its successor’’
Since (1) makes an assertion for arbitrary n it implies
(P)[ ((P(0) & (x)( P(x) P(x|))) (x)( P(x))....(2)
Which is equivalent to a single axiom for mathematical induction.
If, in Peano’s system, we replace the induction rules by (2) we have specified that all
natural numbers shall be successors of zero, and ruled out any non-standard models.
hence every mathematical truth must be true in every model of the revised axiom system,
which is therefore complete.
That may seem like a ‘way round’ Gödel ’s incompleteness theorem, but it isn’t.
Gödel proved that no system adequate for arithmetic can be effectively axiomatisable,
consistent and complete. As the modification of Peano’s system is complete and
adequate for arithmetic it must be either not effectively axiomatisable, or inconsistent. It’s
axioms are satisfiable in arithmetic so it is not inconsistent, hence it is not effectively
axiomatisable - there cannot be any algorithm for determining, of any arbitrary sequence
of formulae, whether or not that sequence is a valid proof. This extension of Peano’s
system is said to be second order because it involves quantification over properties of
numbers (the values of P) rather than just over numbers. Z on the other hand is a first
order system.
This seems the right moment for a short digression on second order logic.
In second order logic it is possible to define equality.
a = b may be defined as (P)(P(a)P(b)), meaning that a and b name the same
individual if all properties that apply to one apply to the other. That implies that if two
names refer to distinct individuals there must be some way of distinguishing them through
some predicate true of one but false of the other. That is what is asserted by the second
order sentence:
a b  (P)(P(a)&~P(b))
Yet what is allowed as a possible value of ‘P’, in the context of that sentence? If we
allow P(x) = (x=a) that seems to make the assertion vacuous, yet there is no clear guide
to what might be permitted.
Another strange second order sentence is one that asserts that any two individuals
have some common property: (a)(b)(P)(P(a)&P(b)) Would a possible value of P be
the property defined by P(x) = [(x=a)(x=b)] ? I’m reminded of a type of joke popular with
small children and the manufacturers of Christmas crackers. Such jokes often take the
form ‘why is a so-and-so like a such and such’ for values of ‘so-and-so’ and
Page 27
What is Philosophy Chapter 4
by Richard Thompson
‘such-and-such’ which do not suggest any striking likeness. For instance ‘Why is an
elephant like a lawnmower?’ answer ‘neither of them is a sewing machine’
For another way of looking at the matter, let V be the set of Gödel numbers of truths.
Then the incompleteness theorem shows that V cannot be defined in arithmetic - there is
no calculation that will determine the members of V. On the other hand the set {V}, of
which V is the only member, can be defined in arithmetic, so that we can at least say
what it is we can’t define. Also, for any value of n, Vn can be defined in arithmetic, where
Vn is the set of Gödel numbers of all mathematical truths than can be expressed with the
use of not more than n of the symbols & , V, , , , , Thus it is possible to have a
system capable of proving all theorems of complexity less than some specified limit.
Categorical Systems
There is another way of approaching attempts to formalise Mathematics. If the aim
is to use a formal system to define the number system, that requires a formal system
which is satisfied by the numbers, but by nothing else. So we should expect to get a
formal system with only one interpretation. Such a system is said to be categorical. A
consequence of Gödel’s incompleteness theorem is that no first order formal system
adequate to express arithmetic is categorical.
However, that is more complicated than it sounds. We need to clarify the concept of
a formal system’s having only one interpretation, since there is one sense in which a
system that has one interpretation must also have others. For instance any system that is
satisfied by the natural numbers 0, 1, 2... and the binary operations ‘+’ and ‘x’ will also be
satisfied by the numbers 0, I, II, III,... and the operations ‘ADD’ and ‘MULTIPLY’, so we
have to rule out alternative interpretations that differ only in notation. The mathematicians’
way of doing that is by using the idea of an isomorphism.
Roughly speaking two models I1 and I2 of a formal system S are isomorphic when
there is a one to one correspondence both between the individuals and between the
functions of I1 and I2, such that if function f1 and individual a1  I1 correspond to function f2
and individual a2  I2 then f1(a1)  I1 corresponds to f2(a2)  I2, and similarly for functions
of more than one variable, and if a sentence S 1 I1 corresponds to a sentence S2 I2 then
S1 and S2 have the same truth value. If all the models of a formal system are isomorphic
the system is said to be categorical.
At this point a complication arises, for in that sense of the word no system with an
infinite model is categorical, since according to the Skölem Löwenheim theorem any
consistent theory with a denumerable model also has a non-denumerable model, and
also any consistent theory with a non denumerable model has a denumerable model too,
but models of different cardinality cannot be isomorphic since by the definition of ‘different
cardinality’ there is no one to one correspondence between them.
Mathematicians therefore settled for the more modest aim that all the denumerable
models of a theory should be isomorphic, and yet even that modest aim that cannot be
achieved by any consistent first order axiomatisation of the number system.
The difficulty is that we want to say that the natural numbers are the numbers that
Page 28
What is Philosophy Chapter 4
by Richard Thompson
can be obtained by starting from zero and counting, 0, 0’, 0’’ ...and those numbers only
but in first order logic we cannot express ‘those numbers only’ As we noted above, to say
that requires (P)[ ((P(0) & (x)( P(x) P(x’))) (x)( P(x)) which involves reference to
‘all properties…‘ and is therefore second order logic.
Incompleteness does not apply to real number arithmetic, only to the arithmetic of
the natural numbers - the notoriously difficult Theory of Numbers, so one might say that
incompleteness is a sign of the inadequacy of the natural numbers as opposed to the
reals. Another way of looking at it is that it shows that the power of a mathematical theory
to express problems is always greater than its power to solve them.
Infinite Sets, Cardinals and Ordinals
Perhaps the oddest part of Mathematics is that dealing with infinite numbers. I shall
outline some of the problems about the infinite as a preliminary to examining the
competing theories concerning the nature of Mathematics.
While Frege was working on the foundations of Mathematics, Cantor was
developing a theory of infinite numbers.
From time to time earlier mathematicians had considered infinite aggregates, but
had been discouraged by apparent contradictions. In the case of infinite numbers there is
even a difficulty about defining equality.
We learn to use ‘equality’ ’greater’ and ’less’ when dealing with finite numbers and
so become accustomed to using two rules. (1) We can establish the number of members
in each of two sets by counting each and comparing the results to show whether their
numbers are equal or unequal. (2) Alternatively we can pair off members of the two sets
and see if any elements are left over. If we are comparing the number of oranges in one
box with the number of lemons in another, and we pair of an orange with a lemon until we
run out of one or the other, and if we find that when we run out of oranges there are still
some lemons left, that shows there are more lemons than oranges.
In the case of infinite sets we cannot simply count their members, because the
process would never end, so we are left pairing, but there is still a problem.
In general terms, finite sets satisfy the following:
(1) S and T have the same number of members if the members of S can be paired
one-one with the members of T
(2a) If S contains every member of T and also some additional elements, then S is
larger than T, for instance (1, 2, 3, 4} is larger than {1, 2, 3}
(2b) If every member of T can be paired one-one with the members of some proper
subset of S, S is larger than T. ( A is a proper subset of S if every member of A belongs to
S, but some members of S do not belong to A, so the set of mice is a proper subset of the
set of mammals)
However in the case of infinite sets (1) and (2) cannot both be true.
Page 29
What is Philosophy Chapter 4
by Richard Thompson
For instance consider the set N = {1,2,3,...} of natural numbers and the set
E = {2,4,6,...} of even numbers.
The function d: n  2n, n  N maps N onto E and
the inverse function h: n  n/2, n  E maps E onto N
so every even number can be paired with a different natural numbers, and every
natural number can be paired with a different even number.
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
...
...
So rule (1) suggests there are just as many even numbers as numbers, while rule
(2) suggests there should be fewer even numbers as they are a proper subset of the
natural numbers..
Cantor proposed to reject rule (2) and specified that two sets are of the same
cardinality if they satisfy (1). All it is possible to save of rule (2) is the weaker rule that, if
A is a subset of S, or can be put in one to one correspondence with a subset of S, then
(cardinality of a)  (cardinality of S)
The failure of rule (2) is one way of defining an infinite set. A set is infinite if its
members can be placed in one-one correspondence with the members of one of its own
subsets. (not, of course, with any one of its own subsets)
A surprising result is that the set of rational numbers (the set containing all the
fractions and whole numbers) has the same cardinality as the set of natural numbers.
That may be proved as follows.
Establishing a one to one correspondence between the members of any set S and
the natural numbers, is equivalent to ordering the members of S. Once ordered they can
be numbered 0, 1, 2,... so we need to order the rational numbers.
Notice first that we cannot order the rationals by arranging them in numerical order.
For supposing we try, starting from zero and working upwards. What rational number
comes next after zero? Not 1/2, for there are lots of positive fractions less than 1/2. For
the same reason not 1/4 nor 1/100 nor 1/1000000.
In general, for any positive fraction p/q, there is a smaller positive fraction, p/(2q)
that is less than p/q but still greater than zero. We must therefore look for another way of
ordering the rational numbers.
For simplicity I shall consider just the positive rational numbers together with zero.
Suppose first that the rational numbers are reduced to their lowest terms so that each is
expressed in the form p/q, where p and q have no common factor greater than 1. The
whole numbers, which are included in the rational numbers, will be represented as
factions with denominator 1, and zero will be represented by 0/1.
Arrange the rationals in order as follows. First group together all those where
p + q = n for each value of n. Arrange the groups in ascending order of the value of n,
Page 30
What is Philosophy Chapter 4
by Richard Thompson
and within each group arrange the numbers in ascending order of size.
so n=1 gives just 0/1
n=2 gives 0/2 and, 1/1 but we delete 0/2 since it is a more complicated equivalent of
0/1
n=3 gives 0/3 (deleted), 1/2, and 2/1
n=4 gives 0/4 (deleted) 1/3, 2/2 (deleted because equal to 1/1) and 3/1
our final list is then: 0/1, 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1...
so that 0/1 is mapped to zero, 1/1 to 1, 1/2 to 2, 2/1 to 3, 1/3 to 4, and so on.
Any set with the same cardinality as the natural numbers is called denumerable or
countable. The members of a denumerable set can be enumerated, that is, they can be
arranged in order with a first, second third... I have just shown that the rational numbers
are denumerable. So are the algebraic numbers.
The real numbers (all the numbers that can be expressed by, finite or infinite,
decimal expansions) are not denumerable. Notice first that, as with the rational numbers,
we certainly can’t count the reals by arranging them in order of size. But that on its own
that does not rule out their being some other way of enumerating them.
Cantor proved that that cannot be done by using his famous diagonal procedure.
His strategy was proof by contradiction. He supposed there is an enumeration of at least
some of the reals, and then showed that if there were it would always be possible to find
a real number that does not appear in that enumeration.
For simplicity consider just the real numbers 0<r<1. Let each number in the
enumeration be written as an infinite decimal. Let the nth real in that enumeration be R n.
We can now construct a number D that is not included in the enumeration. We make
th
th
sure of that by choosing D so that its n decimal place differs from the n decimal place
th
of the n number in the enumeration. There are many ways of doing that; the following is
an example.
Define D as follows. If the digit in the nth decimal place of R n is 7, let the digit in the
nth decimal place of D be 3, otherwise let the digit in the nth decimal place of D be 7. D
therefore differs from every real number in the enumeration.
So if the proposed enumeration begins:
0.156714...
0.07239...
0.22576...
D begins 0.737... The choice of its first three digits ensures that it differs from each
of the first three numbers in the proposed ordering of the reals.
Thus for any enumeration of a set of real numbers there is a real number that is not
included, hence the reals cannot be enumerated. On the other hand the natural numbers
can be mapped onto a subset of the reals by the identity mapping because the natural
Page 31
What is Philosophy Chapter 4
by Richard Thompson
numbers form a subset of the reals. We therefore say that the reals are of greater
cardinality than the natural numbers.
The cardinality of the natural numbers is referred to as  where  is Aleph, the
Hebrew letter A. The cardinality of the reals is called c, for ‘continuum’ because the
number of reals is the number of points on the real number line or continuum. It also
equals the number of points in two dimensions, three dimensional, or any finitely
dimensioned space.
The set of subsets of the realsis of a still higher cardinality called f.
Comparability of cardinal numbers
Notation I refer to the cardinality of set S as n(S), although that is not the standard
notation.
The standard notation is S
but I cannot generate such symbols by using the fonts and styles available in my
word processor and have to create them as graphic elements by using the Microsoft
Equation Editor. That in turn leads to problems in formatting the text so I shall not use that
notation again.
If S can be mapped 1-1 onto a subset of T but not vice versa, we write n(S) < n(T),
and say that T has greater cardinality than S - it has more members. Using the Axiom of
Choice it is possible to prove that for any two sets S and T, precisely one of the following
is true:
n(S) < n(T), n(S) = n(T), n(T) < n(S), so that sets can be simply ordered according to
their cardinality.
It follows that if S can be mapped to a subset of T, either n(S) < n(T), or n(S) = n(T),
so that if S can be mapped to a subset of T and also T can be mapped to a subset of S,
n(S) = n(T), providing a useful indirect way of showing that two sets have the same
cardinality without finding a 1-1 mapping from either of them to the other, so that,
assuming the axiom of choice, the equality of cardinals can thus be established without
actually finding a 1-1 mapping from the whole of one set to the whole of the other.
a
Power sets If a finite set S has a members, the number of subsets of S is 2 > a
That notation is extended to infinite sets. If a set S has cardinality a, the set of subsets of
a
S is called the power set of S and the cardinality of the power set is written 2
a
2 > a holds for infinite cardinals as well as for finite cardinals. The power set of S is
in all cases of higher cardinality than S itself.
a
The argument for 2 > a is as follows. Suppose that there were a 1-1 mapping from
the power set to S. Let St denote the member of the power set that is mapped to the
element t  S So that St is a subset of S.
Now define S* = {x: x  Sx} so S* is the set of elements that do not belong to the
subset with which they are paired
Page 32
What is Philosophy Chapter 4
by Richard Thompson
Note that S* always exists; if there is no x in S satisfying x  Sx, S* is the empty set.
However, S* cannot be mapped to any member of S, since if it were mapped to
some element a,
S* = Sa, and we have the contradiction a Sa  a S*  a  Sa , so the mapping
envisaged is impossible.
There cannot be a set of all sets
Several decades before Russell discovered the paradox about the class of all
classes that do not belong to themselves, Cantor discovered that a paradox arises if we
try to form the set of all sets.
Call that set SS
The SS must contain all its own subsets so its cardinality must be at least as great
as the cardinality of its own power set.
However the cardinality of the power set must be greater than that of SS.
So cardinality (SS) 
Cardinality (2SS)

Cardinality (SS)
Which is a contradiction, so there can be no set SS. ‘All sets’ is at best a proper class.

2 = c = the cardinality of the real numbers. 

To set up a 1-1 correspondence between the set of natural numbers and the set of real
numbers we can restrict our attention to the subset of real numbers 0  x <1, it is easy to
show that that subset can be put in a 1-1 correspondence with the complete set of real
numbers. To do so define y so that y = 0 if x = 0.5, and y = 1/(x - 0.5) if x  0.5, then every
value of x: 0x<1 is mapped to a different real number, and every real number in the set
{x: 0x<1} corresponds to some value of y.
Write the real numbers as bicimals (binary fractions) A real number x ( 0x<1 ) is
then mapped to the set of natural numbers Sx
th
where Sx= {n: the n bicimal digit of x = 1}
Examples: 3/4 = 0.11 B (writing B for binary), so 3/4 corresponds to the set {1, 2)
1/3 = 0.0101010... B so 1/3 corresponds to the set {2,4,6...} = the set of all even
numbers.
Although that procedure maps every real number to a set of natural numbers, and


therefore shows that c  2 , it is not sufficient to show that c  2 because there are
some cases where two bicimals correspond to the same real number and hence two sets
of natural numbers can correspond to the same real number.
There is a one to one correspondence between the sets of natural numbers and the
set of infinite sequences of 1’s and zero’s. However, in some cases two different
sequences of 1’s and zeros, when interpreted as bicimals, correspond to the same real
Page 33
What is Philosophy Chapter 4
by Richard Thompson
number. That happens when a bicimal contains only a finite number of zeros, and
therefore ends in an infinite sequence of 1’s. That infinite terminal sequence of 1’s is
actually a convergent geometric series, and we replace it by its sum.
For instance 0.0100111111.… is equivalent to 0.0101 The numbers in question are
rational numbers with denominator a power of 2
An analogue in the scale of 10 would be 1/9 = 0.11111, so 9/9 = 0.99999..., but we
should actually write that as 1.
Although 0.0100111111.… and 0.0101 represent the same number ( 5/16) they do
not represent the same set of natural numbers. 0.0101 represents {1, 3} but
0.0100111111 represents the infinite set{ 1, 4,5,6.…} containing all the natural numbers
except 0, 2 and 3. It is such sets, those with finite complements, that have so far not been
linked to any real number.
To complete the proof that c  2

we need to show that
2

 c.
That can be achieved by the finding mapping from sets of natural numbers to real
numbers. As there is a one to one mapping from sets of natural numbers to sequences of
1’’s and zero’s, it suffices to find a mapping from those sequences to real numbers, so
that every sequence maps to a different real number.
Let the function u  bicimal(u) map sequences of 1’s and zero’s to the real numbers
they represent when treated as the digits of a bicimal, so that bicimal(1101) =
0.1101(bicimal) = 13/16
Then to map every sequence to a different real, define u  real(u) so that:
real(u) = bicimal(u)/2 if u terminates in an infinite sequence of 1’s,
real(u) = bicimal(u)/2 + ½ otherwise.
So every set of natural numbers can be mapped to a different sequence of 1’s and
zero’s, and every such sequence can be mapped to a different real number, hence 2


c.
As we have shown earlier that c  2

,
2

c
Ordinal Numbers
So far the only infinite numbers I’ve discussed have been infinite cardinals,
numbers that measure the size of a set. Cantor originally studied infinite ordinals, that
measure the length of an ordered sequence of elements. Ordinal numbers correspond to
orderings of a special sort, called well orderings.
To explain that I must discuss order relations generally. I shall use the symbol ‘<’ to
represent an arbitrary order relation though there is no assumption that the ordering is by
Page 34
What is Philosophy Chapter 4
by Richard Thompson
magnitude.
A partial order relation < of S, satisfies: for any x, y, z in S
1. x < y   y < x, the relation is anti-symmetric
2. x < y  x  y , the relation is irreflexive
3. x < y & y < z  x  z, the relation is transitive
Examples are:
people ordered by ancestry, so that ‘a < b’ means ‘a is an ancestor of b’,
propositions ordered by entailment so that P < Q means ‘ P  Q’ ,
sets ordered by inclusion, so that ‘S < T’ means ‘S is a proper subset of T’
A partial order relation may be unable to compare some elements in its domain. For
many pairs of people, neither is an ancestor of the other, and for many pairs of
propositions neither entails the other.
A simple order relation satisfies 1, 2, and 3, and also
4.
(x)( y)(x<y  y < x  x = y)
so that any two distinct elements can be ordered
A well ordering of a set S is defined as simple order relation on S such that every
non-empty subset has a first element. The natural numbers can be well ordered by size,
1, 2, 3..
Ordering the integers by size is a simple ordering but not a well ordering since the
set of all negative integers, when ordered by size, is … -3, -2, -1 which has no first
member. However, the integers can be well ordered, a simple way is 0, -1, +1, -2, +2,...
When they are ordered like that the set of negative integers does have a first member,
namely -1
Well Orderings and Ordinal Numbers
It is well orderings that are measured by ordinal numbers.
All orderings of the members of a finite set are of the same order type and all are
well orderings, since any ordered finite set must have a first member. If a finite set S has
n members, any ordering of S resembles the ordering of the natural numbers {1, 2,..n} so
there is a first member, a second member and, finally an nth member. The orderings of
finite sets are called ordinals of the first class
Not all orderings of an infinite set are well orderings, counter examples are the
orderings according to magnitude of the integers, of the rational numbers or of the real
numbers. However members of an infinite set can be well ordered in many different ways,
each defining an ordinal number (however see the comments below about the Well
Ordering Theorem) Using the set of natural numbers as an example, the simplest well
ordering is:
1,2,3,... which is said to be of order type , the smallest transfinite ordinal
adding an extra element at the end increases the order type, so that
2,3,4,......1 is of order type  + 1 > 
Page 35
What is Philosophy Chapter 4
by Richard Thompson
while adding an additional element at the beginning as in
x, 1,2,3, leaves the order type unchanged, so 1 += 

The concept of adding an additional element at the end of an infinite sequence
needs some explanation. Strictly speaking it is impossible to put an additional element at
the end of an infinite sequence because an infinite sequence has no end. To justify:
2,3,4,......1 we define an order relation x y so that:
x 1 for all x not equal to 1, and provided neither x nor y = 1, x y holds when
x y.
That order relation arranges all the natural numbers except 1 in numerical order, and puts
1 last in the ordering of any set containing 1
I use the symbol ‘‘ to represent the order relation because I want to use ‘<‘ in its usual
sense of ‘less than;
Product of Ordinals The product of two order types, written as , is defined is
the order type obtained by substituting for each element in a well ordering of type  a
well ordering of type 
so  =  since that just represents an infinite sequence of paired elements (1, 2),
(3, 4),... the same as 1,2,3,4.
On the other hand .2 > , since .2 is the ordinal number of the sequence
(1,3,5,.....)(2,4,6,.... In which every odd number comes before any even number.
Once again we have a definition that appears to assume we can finish listing an
infinite set of numbers, and then follow on with other numbers In this case we can define
the relevant order relation thus:
Once again, to save confusion, I use ‘’ to represent the order relation so that I can
use ‘<’ in its standard mathematical sense. Then define a  b as true when
(1) a is odd and b is even
(2) a and b are both even and a<b
(3) a and b are both odd and a<b

It is possible to construct an ordering of type  by substituting a well ordering of
type  in place of each element of another well ordering of type  . One way of doing
that is represented by:
2
3
2
3
2
3
2, 2 , 2 ...3, 3 , 3 ... 5, 5 , 5 .…
In which powers of 2, arranged in numerical order come before powers of 3, and
then successively powers of all the primes.
Page 36
What is Philosophy Chapter 4
by Richard Thompson
Since orderings of finite sets are called ordinals of the first class, and orderings of
denumerable sets are called ordinals of the second class
Although many order types can be defined on a denumerable set it is generally
supposed that we eventually reach order types that require uncountably many elements,
though no one has ever been able to produce an example of such an order type.
*
*
Suppose  is the first ordinal such that any set ordered  is uncountable, then the
cardinal number of such a set is called . In a similar way it is possible to
define,...
Cardinals can be treated as a special class of ordinal, so that a cardinal is defined
as an ordinal u such that there is no ordinal v < u for which n(u) = n(v)
A cardinal can also be defined as a limit ordinal. An ordinal u is the limit of an
increasing sequence 1, 2....r.....of type  if, for all the r and for any  < , all but at
most a finite subset of the r satisfy  < r < .
That shows that  is a limit ordinal, since any ordinal less than  is finite. The
cardinal in question is 
An ordinal that is of the form v = u + 1 where u is another ordinal, is called a
successor ordinal.
Any ordinal is either a limit ordinal or a successor ordinal, so it is either a cardinal or
of the form v+1.
Ordinal Numbers as Sets
An ordinal number can be identified with the set of all lesser ordinals, thus the
ordinal number 3 = {0, 1, 2}. It follows that there can be no set of all ordinals. For suppose
there were such a set and denote it by n, then n defines an ordinal greater than any of
its members, contradicting the assumption that it contains all ordinals. So n can only be
a class, not a set. That is known as the Burali-Forti paradox, one of the logical paradoxes
listed in Principia Mathematica, though it is a paradox only if we assume that there
should be a set of all ordinals.
There is no Greatest Ordinal
There can be no greatest ordinal for suppose there were such a number, and call it
it would be the set of all ordinals less than itself, so that forming a new set by adding 
as an additional member would define a still greater ordinal  + l, contradicting the
assumption that is the greatest. Nevertheless it is customary to use to denote the
‘greatest ordinal’ in informal discussions to decide what new axioms might be adopted as
a basis for extending the theory of the infinite.
The Well Ordering Theorem
According to the Well ordering Theorem every set can be well ordered. The well
Page 37
What is Philosophy Chapter 4
by Richard Thompson
ordering theorem, the axiom of choice and the thesis that all cardinals are comparable
are mutually equivalent in ZF; the axiom of choice could be replaced by either of the
others without affecting the logical power of the system.
The proof of the well ordering theorem is not constructive - it does not involve
showing how to obtain a well ordering of an arbitrary set and so far as I know no set of
cardinality higher than  has ever actually been well ordered, raising doubts about the
well ordering theorem and therefore about the axiom of choice.
Transfinite Induction
A well ordering of a set supports a generalisation of proof by induction, known as
transfinite induction.
Supposing a set S is well ordered with a first element s1 and an order relation <
(which need not be ‘less than’). Then from:
(1) F(s1), and
(2) whenever (x)(x < sF(x)), then F(s)
we may infer (x)(x  S  F(x))

For an ordering of type, transfinite induction is equivalent to simple induction
The Continuum problem
It follows from the well ordering theorem that every infinite cardinal is a limit ordinal,
in other words an Aleph, but the question is which cardinal is which Aleph.

Since c = 2 >  either c = 1 or c > 1The Continuum Problem is the problem of
deciding which Aleph equals c, This cannot be solved in the ZF system. The Continuum
Hypothesis is the hypothesis that c = 
The Continuum Hypothesis can be neither proved or disproved in ZF. That has been
shown by constructing two systems, one by adding the continuum hypothesis to ZF as an
additional axiom, and one by adding its contradictory as an axiom. Each of the resulting
systems is consistent.
n
The Generalised Continuum Hypothesis is the hypothesis that n+1 = 2
Like the Continuum Hypotheses this can be neither be proved nor disproved in ZF.
However if the axiom of choice is replaced by the general continuum hypothesis, the
axiom of choice is then provable, so the hypothesis is in that context stronger than the
axiom of choice.
Large Cardinals
Cardinals are referred to as ‘large’ if they are inaccessible. An inaccessible cardinal
is one that cannot be obtained either
(1) by taking the limit of any set of smaller numbers, nor
a
(2) by an equation of the form b = 2 for any a < b
is inaccessible, because any cardinal t less than  is finite so that 2t is also finite
Page 38
What is Philosophy Chapter 4
by Richard Thompson
and <  but any other inaccessible cardinal  would have to be very large and would
satisfy  =  which, taken in isolation, is strongly counter-intuitive. I gather that
references to large cardinals appear in the proofs of some results in combinetrics.
However it is odd to present that as a reason for accepting large cardinals.
The form of the argument seems to be:
‘We need assumption A to prove theorem T‘.
That assumes we have good reason for expecting T to be true. In other words there
must be reasons R for believing T. Yet R cannot be sufficient to prove T, else there would
be no need to assume A.
Mathematical proofs are often, probably almost always, preceded by informal
discussions and the apparatus of mathematical proof has been developed to replace
such discussions by formal proof. However it seems odd that the wish to formalise the
proof of one result should lead us to adopt an axiom about something else. Perhaps the
informal arguments behind the wish to postulate large cardinals are informal discussions
of such cardinals, yet I have not yet tracked down any non-formal argument that appears
to require large cardinals to formalise it. I should expect to find at least a reference to
some set of large cardinality. Possibly the informal argument needing not be made
respectable is equivalent to something in second order logic which is not clearly valid.
My investigations into the matter are still under way.
A reason for supposing there are large cardinals is the reflection principle, that if
F(), for any conceivable property F, then F(a) for some a < ,where  is the class of
ordinal numbers. The principle is justified by the argument that were F true only of , 
would be an ordinal since it could be defined as ‘the ordinal for which F(  )’. By the same
argument F(a) must be true of  ordinals < for were it true of just  ordinals with  <
, it would be possible to define  as the (+1)th ordinal with property F.
Since an ordinal u is a cardinal if there is no ordinal v < u for which n(u) = n(v) (in
other words if u is not a successor ordinal)must, it is argued, be a cardinal. Since  is
neither a successor ordinal nor a limit of ordinals smaller than itself, it must be
inaccessible, hence there must be  inaccessible ordinals <  . The first of these is 0;
let the second be .
The reasoning of the previous two paragraphs is neither a proof, nor the summary of
a proof, but just an argument for strengthening set theory sufficiently to allow the
existence of inaccessible cardinals to be proved.
It seems to me that the argument is weak because  is not an ordinal, or perhaps it
would be better to say that there is no  so its being false that  is a successor ordinal or
a limit ordinal does not imply that is it an inaccessible cardinal, or any sort of cardinal.
Such inferences are like arguing that, because it is false that the King of France is a Total
Abstainer, he sometimes drinks alcohol.
Theories about Mathematics.
Page 39
What is Philosophy Chapter 4
by Richard Thompson
The various incompleteness theorems make it hard to sustain either Logicism or
Formalism. If no formal system can be complete, it is hard to maintain that such a system
provides a contextual definition of the concepts involved, so that the Logicist can’t be
confident that what he’s constructed in logic really is our ordinary number system, and the
formalist can’t be confident it is the number system that his formal systems have defined.
As we can’t define numbers by means of a formal system, it seems difficult to say
what they are. Platonism, so called because it has some affinity to Plato’s Theory of
Forms, holds that Mathematics is about real abstract objects that exist independently both
of our thoughts and definitions, and of the material world.
Platonism has proved attractive to some workers in the field, notably to Gödel, and
to Russell before the writing of the Principia. But what can it mean to assert that numbers
and other mathematical entities have an independent existence outside space and time?
How could we ever find out anything about such entities? If we can somehow peep into
the world of numbers why do we need to prove their properties instead of using our
insights, and if we can’t peep, how can we tell that our proofs aren’t all completely wrong?
I discuss some of the problems arising from Existence in chapter 5.
Formalism is primarily a strategy for handling formal systems, and if we try to use it
as a theory of mathematical truth, it seems to be the weakest position, since one can
hardly identify Mathematics with a formal system unless there is a unique formal system.
The formalist also has a problem in making sense of consistency proofs. A theory is
pointless unless it is consistent, yet a theory cannot be proved consistent except in a
stronger theory, so with which system should a formalist identify mathematical truth - that
which he normally uses when doing mathematics, or the stronger theory in which he
proves the first theory consistent?
Logicism seems still arguable though only at the cost of including in logic the
informal discussions as to what axioms we may plausibly postulate - we could call that
informal logic, or possibly second order logic, which,
Logicism has been abandoned by almost all Mathematicians and by most
Philosophers, but I think it would be going too far to say that Gödel’s incompleteness
theorem actually refutes Logicism. It was indeed shown that no effectively axiomatisable
system can be adequate for Mathematics, but that does not show that the prospects for
creating mathematical systems are any worse outside Logic than within it. Clearly no
complete system can be created in first order formal logic so we have a number of
systems of variable strength depending on what axioms are adopted.
However the discussion of the merits of various axioms could be regarded as an
application of informal logic. Informal logic must be complete in the weak sense that, for
any mathematical truth T there will be some, more or less plausible, argument for
incorporating it into our mathematics if it is not already a theorem. Second order logic,
which is complete, can even be formalised. But the completeness of second order logic is
not of any practical help in determining precisely which mathematical statements are true,
because in second order logic there is no effective test that can be relied on to establish,
in a finite number of steps, the validity of any proposed proof; but second order Logic may
still have philosophical significance in defining the set of mathematical truths. I don’t,
therefore, think that Logicism can be completely written off, but it seems no more that one
Page 40
What is Philosophy Chapter 4
by Richard Thompson
among several possible ways of looking at Mathematics.
The troubles of Logicism and Formalism encouraged a revival of a more sceptical
tradition with roots going back to Aristotle.
Intuitionism
Before Cantor published his findings it had been generally assumed that there could
be no completed infinity. Aristotle had argued that if there were any infinite collection, we
could consider it divided into parts. For simplicity suppose it divided into just two parts the simplification is mine not Aristotle’s. Then if both parts are finite so is the original
collection. Hence at least one part must be infinite. There is therefore an infinite whole
with an infinite part, contradicting the axiom that the whole is greater than the part.
Aristotle, in common with all thinkers before Cantor, took it for granted that if there were
an infinity there would be only one, so that any infinite collection would be the same size
as any other. He also assumed that a proper subset of any class must have defer
members than the whole class.
Galileo had noticed the one-one correspondence between all the natural numbers
on the one hand and the even numbers on the other, but considered it just evidence of
the paradoxical nature of infinity.
During the first half of the nineteenth century Mathematical Analysis was
developed to free calculus and the theory of series and limits from all references to infinite
collections or to infinitesimals. Cantor then decided to re-examine the status of the
infinite, proposing to avoid Aristotle’s argument by allowing a set to be of the same
cardinality as some of its subsets. After strong initial resistance most mathematicians
acquiesced, but doubts revived as it became apparent that notions of infinity were still
beset by difficulties.
We’ve already remarked on the odd properties attributed to large cardinal numbers,
but closer scrutiny reveals difficulties at an earlier stage, as soon as we go beyond 

The Well Ordering Theorem asserts that every set can be well ordered, yet no set of
cardinality higher than ever has been well ordered. Since ordinal numbers measure
well orderings it follows that no one has ever produced an example of an ordinal number
requiring a set of greater cardinality than  , yet  is defined as the cardinality of the
smallest such ordinal, so that it seems unclear that  so defined actually has any sets to
measure. The problem arises because we have relied on non constructive existence
proofs, to show that there is a well ordering of every set and that there is a set of
cardinality 1 Those proofs which provide no way of constructing the entities whose
existence they purport to prove. The rejection of non-constructive existence proofs was
the rallying cry of a party who came to be known as Intuitionists .
Intuitionism is often considered to have originated with Kronecker (1823-1891)
whose slogan was ‘God created the natural numbers, and man created Mathematics’ but
the name came from Brouwer’s (1881-1966) claim that the basis of Mathematics is
arithmetic, which is in turn based on our intuitive experience of the succession of
moments in time. Intuitionists insist that all existence proofs should be constructive. They
Page 41
What is Philosophy Chapter 4
by Richard Thompson
reject the Axiom of Choice, which postulates the existence of a selection set even if there
is no way of specifying its members, and they deny the existence of any set of cardinality
greater than , understandably since the Axiom of Choice is usually assumed in proofs
that 2

> 
The Intuitionists deny that Logic is prior to Mathematics. They restrict the application
of ordinary logic to finite sets, but assert that the use of infinite sets in Mathematics
requires a different logic. For instance in an infinite domain P V Q may not be a theorem
unless either P is a theorem, or Q is a theorem, so the ‘law of the excluded middle’ ,
P V P is not a theorem.
Heyting (1898-1980) constructed an alternative Intuitionist propositional calculus, with
four connectives none definable in terms of any of the others.
 sometimes construed as ‘it is absurd’ replaces ,
and a full stop replaces &.
The other connectives are V and 
The system is not truth functional, tautologies cannot be identified by constructing
truth tables, instead there is a set of eleven axioms:
(1) P  (P.P)
(7) P  (P V P)
(2) P.Q  (Q.P)
(8) (P V Q)  (Q V P)
(3) (P  Q)  [(P.R ) (Q.R)]
(9) [(P  R) .(Q  R) ] )  [(P V Q)  R
(4) [(P  Q) .(Q  R) ] (P  R
(10)  P (P  Q)
(5) Q  (P  Q)
(11) [(P  Q) . (P  Q) ]   P
(6) [P.(P  Q)]  Q
The rules of inference are the same as those in Principia Mathematica:
substitution: in any axiom or theorem, replace all instances of any individual letter by
any wff, and detachment: if P and P  Q are both theorems, infer Q.
It is hard to decide quite what the various logical terms mean in Intuitionist logic.
When discussing the standard propositional logic we noted that although & V 
corresponded roughly to ‘and’ ‘or’ ‘not’ as we normally use the words, they were not
precisely equivalent and  had only a tenuous connection to ‘if then’ However the
standard logic can be elucidated by truth tables whereas the intuitionist alternative has no
such grounding in truth conditions.
It is particularly odd that the Intuitionist calculus has the theorem  P Pbut
not  P P, suggesting that some propositions are intrinsically negative. It seems to
be permitted to infer ‘Simon is not honest’ from ‘Simon is not not not honest’ but not to
infer ‘Simon is dishonest’ from ‘Simon is not not dishonest’, even though the two
inferences would ordinarily be considered to say the same thing in different words.
The difference between the logical terms, and the roughly equivalent terms in
ordinary language did not matter when we had truth tables to say precisely what the
logical symbols did mean. The Intuitionist has no such way of explaining his terms. Of
course he can explain the system as a whole by saying that the various symbols must
take a set of meanings that make all the axioms of Intuitionist logic true, but that is far
less informative than providing an independent definition for each symbol. Pis not
Page 42
What is Philosophy Chapter 4
by Richard Thompson
equivalent to  P, P V Q is not provable unless either P or Q is provable so that P V
Pis not a theorem. is supposed to approximate to ‘implies’ yet axioms (5) and (10)
seem to provide for a false proposition implying anything and for anything implying a true
proposition, which are the ‘paradoxical’ results that made us withhold the title of ‘entails’
from the truth functional 
Suppose that, within the Intuitionist calculus we define & and  thus:
P & Q is ( P V Q) and P  Q is ( P. Q)
suppose we also treat  so defined as material implication,
then all the tautologies of the classical logic are provable in the Intuitionist system,
which can therefore be viewed not as a subclass, but as a superclass of classical logic,
with additional theorems.
What those additional theorems might amount to became clear when Gödel showed
that Heyting’s calculus is actually equivalent to another system known as Lewis’s S4,
which is one of a number of modal logics constructed by C. I. Lewis (1883 - 1964).
A modal logic is one that includes a symbol corresponding to ‘necessary’ or
‘provable’ Writing  for necessary. Gödel interpreted Heyting’s connectives as:
PP (it is provable that P is not provable)
P.Q = P & Q (P and Q are both provable)
P V Q = P V Q P is provable or Q is provable (the V on the RHS is used in the
truth functional sense)
P  Q = P  Q (the on the RHS is used in the truth functional sense, so that
Heyting’s ‘P Q’ means ‘P is provable materially implies Q is provable)
I discuss modal logic in Chapter 5.
Isaacson
Daniel Isaacson (probably born in the late 1940‘s) defined ‘arithmetical truths’ as the
subset of mathematical truths the truth of which is ‘perceivable directly on the basis of an
articulation of our grasp of the fundamental nature and structure of the natural numbers,
or directly from statements which themselves are arithmetical.’ He held that arithmetical
truths so defined are precisely those that can be deduced from Peano’s axioms. A
putative counter example to that thesis is Goodstein’s Theorem (see the appropriately
titled supplementary document) which concerns finite sequences of natural numbers and
yet is not deducible from the Peano axioms. Isaacson therefore thought himself
committed to arguing that Goodstein’s Theorem is not an arithmetical truth. That is prima
facie most implausible, for Goodstein’s theorem concerns finite sequences of natural
numbers, several of which are short enough to be written down in their entirety.
Supporters of Isaacson make some play with the fact that, except in a very few
cases Goodstein sequences are much too long for us ever to write down the complete
sequence, or even give even a rough estimate of how long the sequence is. However that
amounts to no more than the observation that we cannot carry out the full calculation, and
the same could be said of many results that are decidable within PA - the consequences
of PA must include many theorems so complicated that even the theorem itself is too long
for us ever to state it, let alone carry out a proof. Isaacson is there reduced to saying that
the proof of Goodstein’s theorem, using infinite ordinals, is non arithmetical, but that is a
very dangerous argument for him to use. When defended in that way Isaacson’s thesis is
Page 43
What is Philosophy Chapter 4
by Richard Thompson
easily confused with a different one - that PA encapsulates all our intuitions about the
number system and that for that reason its consequences must comprise the whole
body of arithmetical truth.
Although that was not Isaacson’s argument, it is still worth considering it to see why
it is unacceptable. PA was certainly an attempt to capture our intuitions of arithmetical
truth, and it could at first be plausibly presented in that role because it is closely similar to
a specification that does do justice to those intuitions, namely the second order analogue
of PA. Intuition is an imprecise and fallible instrument, especially when applied to
Mathematics - witness the contradictions in naive set theory, so it cannot be relied upon
to distinguish first order from second order logic, hence the intuitive appeal that appears
to belong to the former may well really arise by confusing it with the latter.
If we allow arguments from the way theorems are proved, separation of the two
theses is not at all easy. Saying that a proof is not part of arithmetic is hard to distinguish
from saying that the proof does not take place within PA, when one is trying to identify
arithmetic with the consequences of PA
As I understand him, Isaacson claims that mathematical truths can be divided into
two classes, lets call them Class A and Class B. He claims that Class A is the set of
arithmetical truths and that all its members are provable in PA, and that class B is the set
of truths are not arithmetical and none of which is provable in PA. If set A were defined
as the set of truths provable in PA, Isaacson's thesis would reduce to: 'Truths provable in
PA are provable in PA'
He therefore needs to distinguish A and B by indicating that truths in the two sets
are somehow qualitatively different, that they make assertions of a different sort so that
they can be distinguished without reference to their proofs.
Isaacson also needs to argue that Gödel sentences are not part of arithmetic. Yet
Gödel sentences take the form of assertions that some diophantine equation, in a lot of
unknowns and of fairly high degree, has no solution. If that is so, those sentences cannot
be located outside arithmetic since, if such an equation does have solutions, that will be
provable in PA since the proof would require only the use of the standard rules for the
multiplication, addition and subtraction of natural numbers. It therefore follows from
Isaacson’s thesis that there are some predicates such that propositions of the form F(a)
are part of arithmetic, yet the corresponding propositions of the form (x)F(x) are not.
Second for some equations with no solutions, their not having solutions will also be
3 4
2
6 2 3
provable in PA . For instance 4y z + 12 x yz - 16 x y z = 5 can have no solutions over
the integers since for any integers x, y, z, the expression on the left hand side must be
even, while the 5 on the right is odd.
Thus it cannot be maintained that the unprovable sentences are of a different sort
from any that are provable. Once again the criterion for being 'arithmetical' seems to be
collapsing into provability in PA, making Isaacson’s thesis and empty tautology.
When I once tried to make this point in conversation it was misunderstood so I’ll
repeat it in slightly different way. I’m trying to say that the equations which cannot be
proved (in PA) to have no solutions are of the same mathematical type as other equations
Page 44
What is Philosophy Chapter 4
by Richard Thompson
some of which do have solutions, and others of which can be proved in PA not to have
any solutions. It is therefore not plausible to say that the Gödel statements are not part of
arithmetic in some sense in which those other sentences that differ from the Gödel
sentences only in being provable in PA, are part of arithmetic. The distinction between the
two classes of sentence is not based on the sort of thing they say, but just on how they
can be proved. Yet Isaacson claimed that the difference in the way statements could be
proved corresponds to a profound difference in the sort of thing they said.
I find it hard to discuss the class of statements provable in PA because in many
cases it is hard to tell precisely what they are. Reference to Mathematical text books is
not helpful because it is customary to use systems stronger than PA for Mathematics, and
the systems mathematicians prefer are not elaborations of PA, but rather prefer
elaborations of Zermelo's system. I don’t know how to equate what Mathematicians use
to something on the lines of 'PA + these additional axioms' and suspect very few people
could do that, because the question would not interest most mathematicians.
Applied Mathematics
When arithmetic is applied to collections of physical objects, we seem to have
propositions that are both logically true, and informative about the physical world. For
instance applying 2 + 3 = 5 to the placing in a box first of two coins, and then of another
three coins, to produce a final total of five coins, seems to give us an arithmetical way of
predicting the consequences of a sequence of physical actions.
However that is an illusion sustained by overlooking two important points. The first is
Frege’s observation that there is no unique number that represents any particular
physical situation. The second point is that in deciding what are suitable objects for
counting, and in choosing physical quantities to be used in measurement, we often favour
types of units and physical quantities that are amenable to arithmetical manipulation.
Consider the following examples:
Example (1) Adding 15 g of sugar to 20 g of water we expect to get 35 g of syrup
(assuming we don’t prolong the process sufficiently for significant evaporation)
o
o
Example (2) Add an egg at 6 C to a saucepan containing water at 100 C, we do
o
not expect to get a combined temperature of 106 C
The scientists’ search for conservation laws is a search for quantities that can be
added in situations where some operation can plausibly be represented as ‘adding’ or
‘combining together‘ objects some feature of which can be measured by those quantities.
Despite the ramblings of some educationists, we don’t investigate the masses of
mixtures to discover the laws of arithmetic. We assume the laws of arithmetic and pick
quantities that exemplify them, regarding mass and energy as therefore more
fundamental than volume, temperature or colour.
In fact it is only in exceptional cases that any physical process at all corresponds to
a calculation.
Example (3) There was £73 in my bank account when I paid in £40. In such a case
the result is defined by the calculation 73 + 40 = 113, and no one would for a moment
Page 45
What is Philosophy Chapter 4
by Richard Thompson
consider the transaction as an experiment to find the answer to the arithmetical
calculation. The only experiment imaginable in those circumstances would be one that
tested the efficiency of the bank’s accounting procedures.
Very often the use of arithmetic corresponds to no process outside the mind.
Example (4) Alexander’s mother has three sisters and his father has two, so
Alexander has 5 aunts. (That ignores inlaws and assumes no incest in his immediate
family)
In a case such we are just finding the number of elements in the union of two
disjoint sets and there is no operation outside our imaginations corresponding to the
formation of the union of those sets. This point is obscured by the fact that in practice we
rarely think of set unions, since the rules for adding natural numbers are implicit in the
counting process, which is commonly employed to introduce children to simple arithmetic.
Discussions of practical arithmetic are sometimes further clouded by discussion of
the consequences of bringing objects of various kinds into close proximity. Some animals
may increase their numbers by reproduction, or reduce their numbers by eating each
other, drops of liquid may merge into one. Although such questions are relevant if we
want to know how many objects will be present in a certain place at a certain time, they
have no bearing on the truth or otherwise of any proposition of arithmetic. It is, indeed,
only by assuming the truth of arithmetic that we are able to tell whether objects placed
together have reproduced or indulged in mutual destruction. If we put five more objects in
a box that originally contained seven, 7+5 = 12 tells us how many objects were put in the
box, and if the number of objects subsequently found in the box differs from 12, the
discrepancy is a sign of foul play, not an indication that arithmetic is doubtful.
In many cases it would not occur to us to expect to find all the objects totalled still
intact. If having eaten two sausage rolls at a party, a guest responds to the host’s appeal
to help finish up the leftovers by eating another three, we calculate the total consumption
as five without entertaining any expectation that five intact sausage rolls repose
somewhere in the guest’s digestive tract.
In a desperate attempt to claim empirical content for the laws of arithmetic, people
have claimed that addition is a generalisation from the results of adding various numbers
of objects of the same sort. In that argument much rests on the phrase ‘same sort’
which either says too little or too much. Any two objects are in some respects the same.
For instance I’ve have heard it said ‘you can’t add two apples to three pears, for what
would be the units of the total?’ Although that makes a point, it does not follow that it is
impossible to apply arithmetic to the aggregation of two apples and three pears; we just
need a suitable notation. In a catering establishment where one apple and one pear each
count as ‘one portion of fruit’ the answer would be ‘five portions of fruit’ On the other hand
if we try to strengthen ‘same’ it is hard to stop short of ‘identical’ so that we never have
collections of things that are all the same.
There is an element of truth behind the claim that we cannot add objects of different
sorts - namely Frege’s point that number is not intrinsic to any object or situation, but only
comes into play when we specify a basis for numbering. Thus ‘2 + 3 = 5’ applies to the
apples and pears only after we have specified that ‘portion of fruit’ is to be the basis of our
counting. However, that does not justify treating the arithmetical statement as an
Page 46
What is Philosophy Chapter 4
by Richard Thompson
empirical generalisation.
Summing up
Mathematics has often been cited as the ideal against which other branches of
knowledge should be measured. It has been represented as a body of certain knowledge
independent of the evidence of the senses. Rationalists have held that Mathematics is a
body of knowledge about a world of ideas, and have cited skill in Mathematics as
evidence that we have some sort of direct access to such a world.
Empiricists have often responded by saying that mathematical truths follow from the
meanings of words used to express them, so that we create mathematical truth when we
define our terms. That empiricist story seems plausible in the case of simple propositions
about the results of individual calculations. ‘2 + 3 = 5’ could plausibly be rendered into
logical notation without using any special mathematical terms at all. (see the appendix 2
where I consider the case of 2+2=4) The same cannot be said of universal statements
about all numbers, but much simple algebra can be deduced from Peano’s axioms, and
those axioms can plausibly be represented as just stating (part of) what we mean by
number.
However, as we progress to stronger systems, such as Zermelo’s set Theory and
the progressively stronger systems obtained by adding additional axioms to that, it
becomes increasingly implausible to regard those axioms as expressions of what we
mean by numbers. But at the same time the successive axioms become increasingly
open to doubt, the doubts becoming stronger with increasing remoteness from our
intuitions about number, so that the Rationalist case for treating Mathematics as a body of
unassailable truth is undermined by the same problems that weaken the Empiricist
counter claim that mathematical truths are truths of logic.
The axiom of choice cannot be represented as true a priori even in Kant’s weak
sense of ‘true in any conceivable world’ because there are Intuitionists eloquently
conceiving its falsehood. The Intuitionists don’t just say ‘we can imagine the Axiom of
choice to be false’ they embody their imaginings in the elaborate detail of an alternative
formal system.
The status of Mathematics is much more complicated and uncertain than was
supposed before the mid twentieth century. I think the balance of argument favours the
Empiricists, since those axioms that go further than expressing the meaning of our
concept of number are not so much based on an intuition of truth as adopted for reasons
of convenience; Mathematicians decide it is convenient to have a particular axiom. If
some Mathematical propositions are not logical truths, they may reasonably be regarded
as the creation of mathematicians. All we can say with certainty is that the precise nature
of Mathematical truth is highly controversial. Mathematics turns out to be a much odder
subject than many had supposed, and its oddities disqualify it from being the template for
any other branch of knowledge.
The philosophical significance of Mathematics is not as great as has often been
supposed.
Technical Appendix: Measurable Cardinals
One motive for wanting large cardinals is measure theory. Measure theory attempts
to formalise such intuitions as that nearly all reals are irrational, because the reals are of
Page 47
What is Philosophy Chapter 4
by Richard Thompson
higher cardinality than the rationals.
Any measure function  defined over a set M satisfies:
 = 0
2. A  B =   (A  B) = (A) + (B)
3. (2) is extended to denumerable collections of pairwise disjoint sets.
(4) A  B  (A)  (B)
1, 2, 3 and 4 do not completely specify a measure function. Adding further axioms
can give any one of a variety of different measures that are used for various purposes.
Lesbegue measure is the basis for a definition of integration that allows an integral
to be defined even in cases where there are denumerably many discontinuities. The
Lesbegue measure of the set of reals a  x  b is b - a. It can be proved that the
Lesbegue measure of any finite set is zero, and so is the Lesbegue measure of any
denumerable set.
For instance, consider Dirichelet’s function d: C  {0,1} where
C = {x: x  R, 0<x<1}
and d is defined by d(x) = 0 if x is rational, d(x) = 1 otherwise,
Lesbegue integration gives
1
 d ( x). dx  1
0
Measurable cardinals arise from the attempt to define a different sort of measure, a
two valued measure over a set M so that, (M) = 1, and (S) is either zero or 1 for any
subset S of M, and in particular (A) = 0 for any finite or denumerable subset A.
To indicate how strange that suggestion is, I have considered the difficulties that
would be encountered if one tried to define a two values measure over the continuum, so
that such a measure, if it could be defined at all, could only be defined over a set of much
greater cardinality than the continuum. That does not show that the set should have
inaccessible cardinality, but I think it does show the oddity of a two valued measure.
The Impossibility of a two valued measure on the Continuum
The requirement that any subset either has measure 1 or measure 0 implies that for
any subset A, and its complement A’ = M - A, one of A, A’ has measure 1 and the other
has measure 1. It also implies that if two subsets have measure 1, their intersection also
has measure 1, for their complements both have measure zero, and so, therefore does
the union of their complements, which is the complement of their intersection. Consider
the set of reals -0.5  x  +0.5 Now consider the subsets
-0.5  x  0 , 0< x  +0.5 and {0} The sum of their measures must equal 1,
m{0} = 0 because the set is finite, so of the two subsets -0.5  x  0 , 0< x  +0.5
one must have measure 1 and the other measure 0.
Pick the one with measure 1 and divide it into two in an analogous way. Continuing
the process we can show that if there where a two valued measure on the continuum
then for any positive integer n, there must be a set of the form:
n
{x: a/2  x  (a+1)/2n} which has measure 1, and the complement of which has
Page 48
What is Philosophy Chapter 4
by Richard Thompson
measure zero which is very odd because as n tends to infinity the Lesbegue measure of
the first set would tend to zero, and that of the second to 1. Of course, the two valued
measure, if it could be defined, would not be the same as the Lesbegue measure, but
such a blatant discordance between two measures would at least be disturbing.
I think the following may actually prove there cannot be a two valued measure on
the continuum. For suppose there were. Let C = {x: 0  x  1, x  R}
Consider L C s.t. x  L  {y:y  x} = 0
clearly 0  L, since {hence L is not empty
Let the least upper bound of L be LB
LB belongs to L, because {y:y  LB} = 0 and { LB} = 0 and the union of two sets of
zero measure itself has zero measure.
Now define the set U  C s.t. x  U  {y:y  x} = 0
clearly 1  U, since {hence U is not empty
let the greatest lower bound of U be UB
UB belongs to U, by an argument similar to that used in the case of L B and L
(I) Suppose LB < UB then (t)(LB  t  UB)
and {y:y  t} = 1 and {y:y  t} which contradicts the properties of a two valued
measure. (The asymmetry in those formulae arises because I need the sets of be disjoint
so that only on of them may contain t, but adding or removing a single element will not
change the measure)
(II) Suppose LB  UB then C = {x: x  UB}  {x: x  LB}, but each of those sets has
measure zero contradicting the hypothesis that C has measure 1
Hence there can be no two valued measure on the continuum.
Theorems about Measurable Cardinals
Preliminary definitions: sup and regular.
If S represents a set of ordinals,
sup(S) = the first ordinal greater than any member of A.
A regular cardinal k is one that cannot be expressed as sup(S) for any set S
containing fewer than k ordinals less than k.
If MC = (a)(a is a measurable cardinal), then
Page 49
What is Philosophy Chapter 4
by Richard Thompson
MC  The General Continuum Hypothesis  The Axiom of Choice
Also MC implies the existence of non-constructible sets including
non-constructible sets of natural numbers and that there are only countably many
constructible sets of natural numbers or of rational numbers.
It is possible to prove that any measurable cardinal M must be inaccessible, and,
assuming the axiom of choice it must be strongly inaccessible, which means that
(1) M is regular and
a
(2) a < M  2 < M
Assuming the General Continuum hypothesis, inaccessibility is equivalent to strong
inaccessibility, so that in the case of a measurable cardinal, whose existence entails the
general continuum hypothesis, the two are equivalent.
Some mathematicians doubt the general continuum hypothesis. Gödel himself
thought that eventually reasons would be found to reject it. If that does happen that would
imply there are no measurable cardinals, and presumably no cardinals even larger than
that.
Supposing some problem required the use a measurable set, what sort of members
might that set have?
The matter seems even more puzzling in the light of the Skölem Löwenheim
theorem according to which every consistent theorem has a denumerable model. Thus
even if a theory contains proofs of the existence of non denumerable sets, that theory can
be interpreted as referring to a universe containing only denumerably many individuals.
Of course denumerably many individuals can be used to construct a non denumerable
collection of sets of individuals, but that does not explain the apparent existence of
inaccessible cardinals which, by hypothesis, cannot be reached by repeated
exponentiation of accessible cardinals. Thus any system that appears to require some set
of measurable cardinality can in fact be satisfied by a model with only denumerably many
elements, and unless measurable cardinals can somehow be conjured up by clever
arrangements of denumerably many elements, they can hardly be required.
Appendix 1
The use of the Axiom of Choice is usually defended pragmatically ‘We need it to
prove so and so....” In that case the theorems cited as unprovable without the Axiom must
have greater plausibility than the axiom itself, suggesting that it is they that should be
axioms. I suspect that the motive for adding the Axiom of Choice is a wish to keep the
system as simple as possible. Axiomatisation is increasingly seen as a tidying up
operation, not as providing mathematics with a basis in a set of self evident truths.
Appendix 2
I think that some simple arithmetical propositions can be reduced to logic. That
certainly seems possible with the sort of examples Philosophers like to use, which are
particular propositions of the form a + b = c, where a, b, and c are natural numbers.
Consider ‘2 + 2 = 4’
Page 50
What is Philosophy Chapter 4
by Richard Thompson
Analyse that as ‘If exactly two objects have property F, and exactly two objects have
property G, and no object has both property F and property G, then exactly four objects
either have property F or have property G’
‘exactly two objects have property F’ is equivalent to
(x)( y)[(F(x) & F(y) & (x = y) & (z)( F(z)  {(z = x) V (z = y)})]
‘exactly two objects have property G’ is equivalent to a similar formula with G in
place of F
‘no object has both property F and property G’ is equivalent to
(x)( F(x) & G(x))
‘exactly four objects either have property F or have property G’ is equivalent to a
formula which I shan’t write out in full, but the following fragments should give a general
idea what is required:
There are elements x, y, z, w, all distinct,
(x)( y)(z)(w)( (x = y) &(x = z) &(x = w) …&(y = z) &…( Z= w) )
such that every one of them is either F or G,
…((F(x) V G(x))& (F(y) V G(y))& (F(z)V G(z) & (F(w) V G(w))
none of them is both F and G,
...&(t)( (F(t) & G(t)), that actually says that nothing at all can be both F and G, but
in the light of the next clause, that has the same truth value as the intended proposition.
and anything that is either F or G, must be one of x, y, z, w
...&(t)( (F(t)VG(t))  ((t = x) V (t=y).....))
Note that nowhere have I quantified over variables representing numbers. I haven’t
said that there is a number 2 or that there is a number 4. This analysis does not assert
the existence of numbers, and they may, for the purposes of evaluating very simple
expressions in arithmetic, be treated adjectivally., and in this context ‘very simple’
includes addition, subtraction, multiplication, or division of any numbers, however large.
Page 51