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§2.1 Models
We tend to think of a set a concrete embodiment of some property. Being blue is
a property of physical objects, so we can talk about the set of blue things. So it’s
tempting to assume that for every property P there’s a set {x | Px} whose elements are
precisely those elements for which the property holds. But this leads to the Russell
Paradox named after the philosopher and mathematician Bertrand Russell. If we can
talk about the set of all sets the surely we can have the set of all sets that are not members
of themselves.
If Ω was the set of all sets then Ω ∈ S. But if Q = “the Queen” then, because she
is not a set, we have Q ∉ Ω. Some sets are members of themselves and others are not.
So it seems obvious that one can have the set of all sets that are not members of
themselves. If Ψ = {x | x ∉ x} then Q ∈ Ψ (since Q ∈ Q is FALSE). And Ω ∉ Ψ
because Ω ∈ Ω is TRUE and hence Ω ∉ Ω is FALSE.
Confused? You’re going to get more confused in a moment, so read on slowly.
We’re going to ask the question “Is Ψ ∈ Ψ?” Our naïve logic insists that either Ψ ∈ Ψ or
Ψ ∉ Ψ. Let’s take each case in turn.
• Suppose that Ψ ∈ Ψ. Then Ψ must satisfy the defining property for Ψ, so Ψ ∉ Ψ.
Clearly this is a contradiction.
• Suppose that Ψ ∉ Ψ. Then Ψ satisfies the defining property for Ψ, and so Ψ ∈
Ψ. Clearly this is also contradiction.
Both alternatives lead to a contradiction and our two-valued logic allows no other
alternative. So we have a contradiction in our mathematics before we even leave the
stage of talking about sets.
Now a contradiction is something that cannot be allowed in mathematics. In
ordinary life we somehow live with contradictions but in mathematics, if just a single
contradiction is allowed in one can prove everything.
Bertrand Russell was once challenged
about this claim. “Assuming that 1 + 1 = 1 prove
that you are the Pope,” he was asked. Russell
gave an argument along the following lines:
Suppose that 1 + 1 = 1.
Now by definition, 1 + 1 = 2.
Therefore 1 = 2.
The Pope and I are two people.
Therefore the Pope and I are one person.
Therefore I am the Pope!
In the nineteenth and early twentieth centuries mathematicians were concerned
with the foundations of the subject, and philosophers were concerned with the nature of
truth. They developed mathematics on the basis of set theory. There was basically only
one “axiom” about sets that needed to be used to create this mighty edifice, though it was
never stated explicitly.
Axiom of Extensionality: For every property P there is a set that consists of all sets that
have that property. In symbols: {x | Px} is always a set.
The empty set is a set because it’s {x | x ≠ x}.
{a, b} is a set because it’s {x | x = a or x = b}
We can define x+ = {x, {x}} and hence we can define the integers by considering n+ as
n + 1 (though addition and multiplication would yet have to be defined).
In the early 1900s as the great philosopher Frege was preparing the second
volume of his book on the foundations of mathematics, building everything on the basis
of the axiom of extensionality. But just before it was published Bertrand Russell wrote to
him pointing out what we now know as Russell’s Paradox.
The contradiction that arises from this paradox shows that the foundation that
underpinned Frege’s book was invalid. The book had to be withdrawn from publication.
Mathematics was in danger of collapsing! A few mathematicians, those interested in the
foundations of mathematics, tried to prop it up. Most mathematicians simply ignored the
problem and just got on with their business.
The rescue came with replacing the one axiom by a set of axioms that avoided the
Russell Paradox. Several axiom systems have been proposed, but they have all been
shown to be equivalent to all the others. One of the most widely used sets of axioms is
the Zermelo-Fraenkel axioms. This allows {x | Px} to be a set only for certain specified
properties and these will avoid the property x ∉ x.
We define a model to be a collection of objects, called sets, together with a binary
relation ∈ such that:
(Axiom of Equality): ∀S∀T[∀x[x ∈S ↔ x ∈T] ↔(S = T)]
If x ∈ S we say that “x is an element of S”, or “x is a member of S”, or “S contains x”.
But empty your mind of any intuitive notion you may have of membership.
So in a model two “sets” have to be equal if and only if they have the same
“elements”. If you think this is obvious you are relying on your intuition and knowledge
of the usual meaning of the word “member” or “element”. Consider the following
Example 1: Let’s take the system ℕ of natural numbers and define membership by:
x ∈ y ↔ x2 < y.
Then 2 is an “element” of 9 because 22 < 9. Using the { } notation for listing elements
we have 9 = {0, 1, 2} because these are the only natural numbers whose square is less
than 9. But 8 = {0, 1, 2} as well. But 8 ≠ 0 so this violates the axiom of equality. This
means that this example is not a model.
Example 2: Let’s take the collection of positive integers but this time we’ll define ∈
slightly differently, by
x ∈ y ↔ x < y2.
In this model 9 = {1, 2}. More generally, n = {1, 2, ..., n2 − 1}.
This system does satisfy the axiom of equality and so is a model.
There’s a philosophical difficulty in that we’ve used the word “collection” on the
way to defining sets. What exactly is the difference between a set and a collection? It’s
hard to get anywhere in mathematics without somehow invoking the naïve concept of a
set. Logicians face the same problem when discussing logic. They’re forced to use logic
in the process of talking about logic.
Let’s just say that the word “collection” refers to our intuitive idea of things
belonging. When we talk of “sets” we attempt to empty our mind of the ordinary
meaning of a set and to assume only what is set out in our axioms. In the following
example, our sets are things not normally thought of as collections.
If P is a predicate built up from ∈ and there exists, in the model, a set whose
elements are precisely those sets x for which Px is true then we write that set as {x | Px}.
We don’t assume that {x | Px} is always a set. If it is not a set we call it a proper class.
So in the set theory that we will develop, {x | x ∉ x} will be a proper class.
One could ask where proper classes live, if they’re not sets in our model, or
whether they even exist anywhere. Similar questions no longer bother mathematicians
about imaginary numbers. If we say they exist then they exist. What is important is not
existence, but how they behave.
We shall assume that for all predicates P, {x | Px} exists – where doesn’t matter.
We call these imaginary objects “classes”. If a class corresponds to a set within our
model then it is also a set. Otherwise it’s called a proper class.
If a set contains no elements it is called an empty set. By the Axiom of Equality,
all empty sets are equal, so in a model there is at most one empty set. If such a set exists
the it is denoted by the symbol ∅.
Example 3:In the model of Example 2, 1 = ∅, the empty set.
If we modify the definition of membership as follows:
x ∈ y ↔ x ≤ y2
we get another model. But in this model we have no empty set because every set
contains the element 1.
Example 4: Let’s revert to the model of Example 2. That is, x ∈ y means that x < y2.
The “sets” are:
1 = {}, the empty set,
2 = {1, 2, 3},
3 = {1, 2, 3, 4, 5, 6, 7, 8} etc.
Then {x | (x ∈ 2) ∧ (x ∈ 3)} = {1, 2, 3} = 2. Since this is in the model it is a set.
But {x | (x ∈ 2) ∨ (x = 5)} = {1, 2, 3, 5}. Since this is not in the model it is a proper
For any model we define the following classes. They may or may not be sets.
The extension {x | Px}, where y ∈ {x | Px} ↔ Py. [P is any predicate built up from ∈.]
A particular case is {x ∈ S | Px}, which means {x | Px ∧ (x ∈ S)}.
The difference S − T = {x | (x ∈ S) ∧ (x ∉ T} (the difference of two sets).
The empty set is defined to be {x | x ≠ x} and is denoted by ∅.
The unordered pair {S, T} = {x | (x = S) ∨ (x = T)}. More generally {x1, x2, … , xn}
denotes {x | (x = x1) ∨ (x = x2) ∨ … ∨ (x = xn)}.
The union of a set S is {x | ∃y[x ∈ y ∧ y ∈ S} and is denoted by ∪S.
The intersection of a non-empty set is {x | ∀y[y ∈ S → x ∈ y]} and is denoted by ∩S.
You should be familiar with the union and intersection of two sets, but ∪S and
∩S are something new. But these definitions are just an extension of what you know.
These allow us to talk about the intersection and union of any set of sets.
The intersection of two sets S and T is defined to be S ∩ T = ∩{S, T}, assuming that
{S, T} is a set. We say that S, T are disjoint if S ∩ T = ∅.
The union of two sets S and T is defined to be S ∪ T = ∪{S, T}, assuming that {S, T} is
a set. If S, T are disjoint we often write S ∪ T as S + T.
Now mathematics contains many more concepts than sets and elements of sets.
We have ordered pairs, and integers, and real and complex numbers. There are functions,
and matrices and geometric objects such as triangles and circles. Our goal will be to
define all of these in terms of the relation ∈. In this way we can build up all of
mathematics within a certain model. Let’s begin with ordered pairs.
How do you define (x, y)? You might say that it consists of two things in a
certain order – except that (x, x) is an ordered pair so there might only be one thing. You
know intuitively what (x, y) means but giving a precise definition would appear to be
tricky. Here’s how we do it.
The ordered pair (x, y) is defined to be {{x}, {x, y}}.
This may seem a strange way of defining an ordered pair but it has the one important
property that we expect from an ordered pair, namely that (a, b) = (c, d) if and only if
a = c and b = d.
Theorem 1: If u = (a, b) and v = (c, d) then u = v implies that a = c and b = d.
Proof: ∩u = {a} ∩ {a, b} = {a} and so ∪∩u = a.
Similarly ∩v = c. Since u = v, we have a = b.
Also ∪u = {a, b} and so ∪u − ∩u = {c}. Hence ∪(∪u − ∩u) = b.
Similarly ∪(∪v − ∩v) = d. Since u = v, we have b = d.
The cartesian product S × T is defined to be {(x, y) | (x ∈ S) ∧ (y ∈ T}.
S is a subset of T if: ∀x[x ∈ S → x ∈ T]. We denote this by writing S ⊆ T. So S = T if
and only if S ⊆ T and T ⊆ S.
The power set ℘(S) is defined to be {x | x ⊆ S} = {x | y ∈x → y ∈ S}.
Notice how all of these constructions can be written entirely in terms of ∈.
S+ = {x | (x ∈ S) ∨ (x = S)} called the successor of S.
In other words S+ = S ∪ {S}.
A set x is called a successor set if it contains the successor of each of its elements.
The successor closure of S is the intersection of all the successor sets that contain S (as
an element). We denote it by S*.
So S* = ∩{x | (S ∈ x) ∧ ∀y(y ∈ x → y+ ∈ x)}.
These last two constructions are used in the construction of the natural numbers. We will
be defining 0 to be the empty set and for every natural number n, we will define n + 1 to
be n+. A successor set that contains 0 must therefore contain all natural numbers. The set
of natural numbers will be defined to be 0*, the smallest successor set that contains 0.
Example 5: Consider the model in Example 2, consisting of positive integers where
x ∈ y means that x < y2.
1 = {}, the empty set,
2 = {1, 2, 3},
3 = {1, 2, 3, ..., 8},
4 = {1, 2, 3, ..., 15},
3 – 4 would be the set of all integers x less than 32, but not less than 42. There are no
such numbers so 3 − 4 = 1, the empty set. On the other hand 4 − 3 would consist of all
numbers that are less than 42 but not less than 32, that is, {9, 10, ..., 15}. But this is not a
set in this model and so it is a proper class.
∪3 = {x | ∃y[(x < y2) ∧ (y < 32)} = {1, 2, 3, ... 63} = 8.
More generally ∪n = n2 − 1, except for ∪1 which is 1, the empty set.
For all n, ∩n = 1.
3+ = {x | (x ∈ 3) ∨ (x = 3)} = {0, 1, 2, 3, 4, 5, 6, 7, 8} = 3.
More generally n+ = n.
Remember that we are going to define n + 1 to be n+. Clearly if we tried to do this within
this model we’d have an arithmetic that was quite different to the one we know.
(1, 2) = {1, {1}} is a proper class in this model because {1} is not a set.
2 ⊆ 3 because the elements of 2 are all elements of 3.
℘3 = {1, 2, 3} = 2 is a set but ℘4 = {1, 2, 3, 4} is a proper class.
This rather strange example is designed simply to demonstrate the fact that the
axioms for sets make no assumptions as to what the objects are and require no intuitive
concept of a set as a collection. In fact intuition just gets in the way. One has to simply
go back to the definition of membership all the time.
One novel feature of this model is that the integers from 2 onwards are elements
of themselves. The phenomenon of x ∈ x is an interesting one. When we come to
setting up the axioms for set theory we’ll have to decide whether to allow this possibility
or whether to rule it out. Before you reach that point in the notes you might like to
contemplate whether you would like to allow this self-referential behaviour of sets.
§2.2 ZF-Models
A ZF-model is a model that satisfies the following axioms:
(1) Empty Set: ∅ is a set.
(2) Pairs: If S, T are sets so is {S, T}.
(3) Powers: If S is a set so is℘S.
(4) Union: If S is a set so is ∪S.
(5) Infinity: ω = 0* is a set.
(6) Specification: If S is a set and P is any predicate
built up from ∈ then {x ∈ S | Px} is a set.
(7) Substitution: If S is a set and F is any function then
F[S] = {F(x) | x ∈ S} is a set.
We will be defining functions later as sets of
ordered pairs. For the purpose of Axiom (7) a function
is a binary predicate Pxy, built up from ∈, such that
∀x∀y∀z[Pxy ∧ Pxz → y = z].
Without the The Axiom of the Empty Set we
would have no sets at all. But this axiom on its own
only produces one set, ∅.
By the Axiom of Pairing, if S is any set then so
is {S, S} = {S}. So with just the first two axioms we
can produce ∅, {∅}, {{∅}}, … and pairs of these,
such as {{∅}, {{{∅}}}}.
We can produce infinitely many others, such as {{∅}, {∅, {∅}}}. But all these
have 0, 1 or 2 elements.
If we take just the first three axioms we can produce larger sets.
For example ℘(∅) = {∅} with 1 element,
℘2(∅) = {∅, {∅}} with 2 elements,
℘3∅ = {∅, {∅}, {{∅}}, {∅, {∅}} with 4 elements
But all such sets will have size 2n for some n.
With the first four axioms we can get sets of any finite size. For example:
{∅, {∅} ∪ {∅, {{∅}} = {∅, {∅}, {{ ∅}}}. But all such sets will be finite.
We need the Axiom of Infinity to get an infinite set and with the Axiom of Specification
we can be sure that subclasses of sets are indeed subsets. The Axiom of Substitution is
rather more technical than the others, but in essence it says that any class that is in 1-1
correspondence with s set is a set.
We’ll make a lot of use of the Axiom of Specification. For any property P and
any set S there is a set whose elements consist of those elements of S that satisfy P. The
property has to be something that’s built up from primitive statements of the form x ∈ y.
This is a restricted form of the one that led to Russell’s Paradox.
We won’t be able to consider {x | x ∉ x} as a set. The best we could do is to
consider Ψ = {x ∈ S | x ∉ x} for some set S.
If Ψ ∈ Ψ then Ψ ∈ S and Ψ ∉ Ψ. This is obviously a contradiction.
If Ψ ∉ Ψ then all we can say is that either Ψ ∉ S or Ψ ∈ Ψ. The latter is a
contradiction but the former is a possibility that avoids a complete contradiction. Hence
we have proved that Ψ ∉ Ψ and Ψ ∉ S. So what! This is no contradiction.
But reflect again on the fact that in order for the Axiom of Specification to work
we need to have sets in which to operate. And without the Axiom of the Empty Set our
model could be empty. The Big Bang that creates the infinite universe of sets from a
void is the axiom that assumes the existence of the empty set.
There’s something rather appropriate about mathematics being created out of the
empty set. If you have a religious bent you can liken it to God creating the world out of
nothing. If you have a scientific bent you can liken the process to the Big Bang, which
sort of says the same thing, without religious overtones.
We assume the existence of a ZF-model. This will be our universe of sets within
which all of mathematics can be developed. So, from now on we work within a ZF
model. Whether such a model exists is another matter. These axioms have never been
proved to be consistent, and probably never will be, because to prove consistency we’d
have to create a model that satisfies them, and we can only do this by somehow using our
ZF model. All we can do is to prove theorems based on these axioms and hope for the
The ZF axioms are really a creed. Virtually all mathematicians consciously, or
unconsciously, believe in this creed, or something equivalent to it. There are a few
agnostics who deny the axiom of infinity on the grounds that we live in a finite universe.
But they are the losers. Their mathematics is severely impoverished.
Just like a religious creed we cannot prove the ZF axioms to be true. What would
we start with in order to do this? At least most religious creeds can claim to be
consistent, which is more than seems to be the case with the ZF axioms. These have
never been proved consistent – but then they have never been proved inconsistent. We
haven’t even got a proof that it is a logical impossibility to prove them consistent, though
most mathematicians believe this to be the case.
So here is the point where you can give up mathematics altogether and go and do
gardening or something else. If you want to be a serious mathematician and want to base
your mathematics on a firm foundation, I’m sorry, the ZF axioms, or their equivalent is
the best we’ve got. If one day someone comes up with a new paradox that shows the ZF
axioms to be inconsistent, a few mathematicians will undertake the job of modifying the
fundamental axioms, while the vast majority will continue as if nothing has happened! In
the mean time we now show that certain familiar mathematical objects that we are
familiar are indeed sets.
Theorem 2: For all sets S and T, S − T is a set.
Proof: S − T = {x ∈ S | x ∉ T} is a set by the Axiom of Specification.
Theorem 3: ∩S is a set if S ≠ 0.
Proof: Let T ∈ S.
Then ∩S = {x ∈ T | ∀y[y ∈ S → x ∈ y]} by the Axiom of Specification.
Theorem 4: For all sets S and T, S ∪ T is a set.
Proof: By the Axiom of Pairs, {S, T} is a set.
By the Axiom of Unions, ∪{S, T} = S ∪ T is a set.
Theorem 5: For all sets S, T, (S, T) is a set.
Proof: By the Axiom of Pairs {S, T} is a set. By the Axiom of Powers ℘{S, T} is a set.
Now (S, T) = {{S}, {S, T}} is clearly a subset of ℘{S, T} by the Axiom of Specification.
Theorem 6: For all sets S and T, S × T is a set.
Proof: If x ∈ S and y ∈ T then (x, y) ⊆ ℘{x, y} and so (x, y) ∈ ℘2{x, y}. Now each of x
and y is an element of S ∪ T, which is a set by theorem 5. So {x, y} ⊆ S ∪ T and so
(x, y) ∈ ℘2(S ∪ T), which is a set by the Axiom of Powers.
Hence S × T ⊆ ℘2(S ∪ T) and so is a set by the Axiom of Specification.
We can write out explicitly the property that extracts S × T from all the other
elements of ℘3(S ∪ T) as follows:
S × T = {z ∈ ℘2(S ∪ T) | ∪∩z ∈ S and ∪(∪z − ∩z) ∈ T}.
To express the condition in terms of primitive membership statements is
straightforward, but very messy. For a start, if y = {x ∈ z | Px} is a set we can express
y ∈ S by ∃s[s ∈ S ∧ ∀x[x ∈ s ↔ Px]].
Now x ∈ ∩z can be written as ∀y[y ∈ z → x ∈ y] and x ∈ ∪z can be written as
∃y[x ∈ y ∧ y ∈ z].
So ∪∩z ∈ S can be expressed as ∃s[s ∈ S ∧ ∀x[x∈s ↔ ∃a[x∈a ∧ a∈∩z]]],
that is ∃s[s∈S ∧ ∀x[x∈s ↔ ∃a[x∈a ∧ ∀b[b∈z → a∈b]]]].
And ∪(∪z − ∩z) ∈ T can be expressed as ∃t[t∈T ∧ ∀x[x∈t ↔ ∃a[x∈a ∧ a ∈ ∪z − ∩z]]].
that is ∃t[t∈T ∧ ∀x[x∈t ↔ ∃a[x∈a ∧ ∃b[a∈b ∧ b∈z] ∧ −∀b[b∈z → a∈b]]]]
that is ∃t[t∈T ∧ ∀x[x∈t ↔ ∃a[x∈a ∧ ∃b[a∈b ∧ b∈ z] ∧ ∃b[b ∈ z∧ −[a∈ b]]]]].
I think you get the idea. Now we don’t want to have to crawl at this basic level all
the way through our development of mathematics. The point is that we could do so, if we
really had to.
We can’t get very far with mathematics without functions and relations. One of
the very first things we learnt in arithmetic was 2 + 2 = 4. Before we can justify this
rigorously we need not only to define the numbers 2 and 4, but also addition, which is a
function of two variables.