Download Appendix A Infinite Sets

Appendix A Infinite Sets:
Part I: What Does Infinity Mean?
So what does infinite really mean? We use the word regularly, but how do we define it?
Generally, people would define infinite as "endless", "without beginning or end", “going on
forever” (which begs the question, “what does forever mean?”) or “going on and on”. Another
response is the somewhat self-referencing, “anything that is not finite”. None of these
definitions are rigorous, and in mathematics we usually want rigor. First, we start by
recognizing that infinite is an adjective, it is a characteristic that something may have; thus
rigorous definitions of infinite in mathematics generally depend on what the adjective infinite is
modifying. If you take a Calculus or analysis course you will, most probably, define infinite
limits. In this appendix we will define infinite sets.
There are two ways to go about defining infinite sets, one easier for most people to
follow than the other. We will begin with that one. This approach is to first rigorously define
what we mean by finite sets then define infinite sets as sets that are not finite. Let us start with
our intuitive idea of a finite set. If a set is finite, then, given enough time, theoretically, we could
count its elements. (I say theoretically because, while I believe that the set of grains of sand on
the planet is finite, I do not believe it is humanly possible to count them.) So what do we mean
when we say, for example, this set has 10 elements? What exactly are we doing when we count?
Imagine for a moment that you are a member of an ancient civilization, one that has not
yet developed a counting system. Further imagine that you are a shepherd with a collection of
sheep and you want to take them out in the countryside to graze then bring them all back home.
How would you be sure that they all came back with you, that you didn’t lose any along the
way? If your collection of sheep is small enough, you can get to know each sheep personally,
give them each a name. Then you might notice that little Sally Ewe must have wandered off
because you don’t see her anywhere. But then you need to remember all the names. You might
try the buddy system we use with children but sheep don’t usually cooperate or communicate
well enough for that. Also, if you were missing an entire pair, how would you know?
Basically, you need a way to count them, but without using numbers. One way to do so,
if you have the natural resources available, would be to take a large collection of stones and
make the sheep pass through a gate one at a time. As each sheep passes through the gate you
would place a stone in a box or bowl you have constructed for this purpose. (Ok, so I did use the
number one but there is evidence that early civilizations had words for one, two and many before
they had developed number systems.) When you return with your sheep, you can again have
them pass through the gate taking a stone out of your box or bowl for each sheep. If, when they
have all passed through the gate, you have any stones left in your box/bowl, then you have lost a
sheep for each remaining stone. Otherwise, you’ve brought them all back. For this discussion
we will ignore questions about how you convince the sheep to pass through the gate in an
orderly fashion, I'm sure you could come up with something along these lines that would work;
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after all you wouldn't be reading a discrete mathematics text if you weren't a problem solver.
Now suppose you realize one day that the stone business is rather tedious and besides
which, if you want to take your sheep further afield and check that you have them along the way,
the stones are rather heavy to carry around. So you decide to invent some special words, which
you memorize to represent the stones. You always use the words in exactly the same order so
you know which words you have used, just by remembering the last one. Instead of putting the
sheep in one-to-one correspondence with stones, you put them in one-to-one correspondence
with the words. Now you’re cooking, I mean, counting.
Let us now return to the present day where we have a well defined set of “counting
numbers”, {1, 2, 3, . . . } and we can make the following definitions:
Definition A.1 A one-to-one correspondence between the elements of two sets is
a matching where each element of one set is matched with exactly one element of
the other where no elements of either set are left out. (note: if you have read
section 1.3, you could write a cleaner definition using one-to-one and onto
functions)
Definition A.2 A set S whose elements can be put in one to one correspondence
with the elements of the set {1, 2, . . . ., n} of counting numbers is said to have n
elements or to have cardinality n.
We write |S| = n.
Definition A.3 Two sets A and B are the same size or have the same cardinality
if there exists a one-to-one correspondence between their elements.
Now we have a relatively straight forward definition of finite:
Definition A.4 A set is finite if it has n elements for some counting number n.
Now it is perfectly rigorous to say:
Definition A.5 A set S is infinite if it is not finite.
So are all infinite sets the same size? First, consider the set N of all the counting numbers. Is it
infinite? How do you prove it?
Theorem A.1: The set N of counting numbers is infinite.
We will prove this by contradiction. See section 2.0 and 2.0 problem 3 for a discussion of this
method of proof. Before we begin, however, let us note that if there is a correspondence
between the elements of {1, 2, . . . ., n} and the elements of a set of distinct numbers K, it is
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possible to find the largest element of k using the following algorithm:
•
•
•
•
•
•
Algorithm for finding the greatest element in K:
Let L be the number corresponding to 1.
Let I=1
(*)If the number corresponding to I+1 is greater than L then let L be that number.
Let I=I+1
If I # n then go to (*).
When the algorithm terminates, L is the greatest element of K.
”
Now we can prove theorem A.1:
Proof:
Suppose N is finite.
Then there is a counting number n such that there is a correspondence between the
elements of {1, 2, 3, . . ., n} and N.
Then there is a greatest element L of N which can be found using the above algorithm.
Since L is in N, L is a counting number. So L+1 is a counting number greater than L. This is a
contradiction to the fact that L is the greatest element in N. Therefore our initial supposition
must be false and the theorem is proved by contradiction.
”
Of course, you may say that it is obvious that N is infinite. Generally things that are
obvious are the most difficult to prove. Sometimes being completely rigorous in mathematics is
a little like dealing with a two-year-old in the “why” stage. You want to say “proof: because I
said so!” but you have to bite your tongue.
Now let us consider the set of even counting numbers, E = {2, 4, 6, . . .}; is it the same
size as N? Instinct may have you saying “no” since every element of E is in N but not the other
way around. Clearly this says there are more elements in N? We have an obvious
correspondence (not one-to-one) which pairs everything in E with something in N leaving
nothing in E out but leaving infinitely many elements of N out. But beware! Just because one
correspondence doesn’t work doesn’t mean there isn’t one that does. Consider the following
mapping:
1->2
2->4
3->6
4->8
!
n->2n
!
matching every element of N with its double in E. This is a one-to-one correspondence between
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N and E. (Verification is left as an exercise.) So E and N are the same size, even though E is a
proper subset of N! (For a discussion of proper subsets see section 1.0.) You can’t do that with
finite sets! (try) which leads to an alternate definition for infinite:
Definition A.4(alternate) A set S is infinite if it can be put in one-to-one
correspondence with a proper subset of itself.
And for finite:
Definition A.2 (alternate) A set S is finite if it is not infinite.
It is left as a challenge to the reader to prove1 that the two pairs of definitions (our originals and
the alternates) are equivalent.
Now let’s return to our earlier question, “are all infinite sets the same size?” A way to
rephrase this would be: “can all infinite sets be put in one-to-one correspondence with the set of
counting numbers?” Well, any set whose elements can be listed in order, say from largest to
smallest or any other linear order could since you could correspond the first element in the list
with 1, the second with 2, etc. So, if there is an infinite set that cannot be put in one-to-one
correspondence with the counting numbers then it must be one that cannot be put in order from
largest to smallest. The integers don’t have a smallest element but they can be listed if you start
at 0 and alternate between positive and negative: 0, -1, 1, -2, 2, -3, 3, . . .. Let’s consider the set
of all positive fractions of counting numbers; they can’t be put in order from smallest to largest
either, so you might be tempted to conclude that there are more of them than there are counting
numbers. But remember, just because one method of listing them won’t work, don’t assume
none will.
Consider the following doubly infinite array of fractions:
1/1
2/1
3/1
4/1
5/1
6/1
...
1/2
2/2
3/2
4/2
5/2
6/2
...
1/3
2/3
3/3
4/3
5/3
6/3
...
1/4
2/4
3/4
4/4
5/4
6/4
...
1/5
2/5
3/5
4/5
5/5
6/5
...
1/6
2/6
3/6
4/6
5/6
6/6
...
!
!
!
!
!
!
"
1
A proof of this requires choosing an element from the infinite set. There is
actually an axiom in mathematics called the Axiom of Choice which allows us to do this.
If one does not accept the axiom of choice then this equivalence would not necessarily
hold.
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Each positive rational number will appear in the array (in fact it will appear infinitely
many times). Now, notice (see figure 1) that you can draw a path through the array, which, if
continued, will visit each rational number, one at a time. When you reach an entry that is not in
reduced form (circled below), simply skip it and pass to the next. The numbers you visit, in
order along this path provide list of the rational numbers and a blueprint for putting them in one
to one correspondence with the counting numbers.
1/1
2/1
3/1
4/1
5/1
6/1
.. .
1/2
2/2
3/2
4/2
5/2
6/2
.. .
1/3
2/3
3/3
4/3
5/3
6/3
.. .
1/4
2/4
3/4
4/4
5/4
6/4
.. .
1/5
2/5
3/5
4/5
5/5
6/5
.. .
1/6
2/6
3/6
4/6
5/6
6/6
.. .
!
!
!
!
!
!
"
The list this provides begins: 1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, . . .and will
contain all positive rational numbers exactly once.
For ease of discussion, let us define a term for sets that can be put in one-to-one correspondence
with the counting numbers:
Definition A.6: Any set that can be put in one-to-one correspondence with
a subset of the natural numbers is said to be countable. Infinite sets that
can be put in one-to-one correspondence with the set of natural numbers
are said to be countably infinite or denumerable.
So all finite sets are countable; the integers are countable; and the rational numbers are
countable. Are there any sets that are not countable? This discussion would not be very
interesting if the answer were no, so you have probably guessed that there are. Consider the set
of real numbers in the interval [0,1] that is, real numbers x such that 0#x#1. At the end of the
19th century a controversial mathematician named Georg Cantor (1845-1918) developed a
diagonal process (the Cantor Diagonal Process) to show that this set is not denumerable.
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Cantor's was interested in medieval theology2, particularly its arguments involving the
continuous and the infinite, this led him to develop a theory of infinite sets which resulted in
some disagreement in mathematical philosophy. His biggest critic was Leopold Kronecker after
whom the Kronecker delta "function", another controversial tool used extensively by physicists,
was named. In any case, Cantor's diagonal process provides a now widely accepted method of
proving the following:
Theorem A.2: The set of real numbers between 0 and 1 is not countable.
Proof (using the Cantor Diagonal Process):
Suppose that this set were denumerable, then there would be an infinite list of real
numbers between 0 and 1 which would contain them all. But, we can show that, given any list of
these numbers, there is an algorithm for constructing a number not in the list. The algorithm
relies on the fact that every real number between 0 and 1 has a decimal expansion, all of whose
non-zero digits lie to the right of the decimal point and that every non-zero decimal expansion all
of whose non-zero digits lie to the right of the decimal point is a representation for a real number
between 0 and 1. Furthermore, if we do not allow expansions ending in an infinite string of
nines, this representation is unique. Here is the algorithm:
Algorithm for constructing a real number d, not in a given list of real numbers:
Let d be the number whose decimal expansion is given by: 0.d1d2d3 . . . where the digits di
are defined as follows: If the first digit to the right of the decimal point of the first number in the
list is a 1 then d1 = 2, otherwise d1 = 1. If the second digit to the right of the decimal point of the
second number in the list is a 1 then d2 = 2, otherwise d2 = 1. And so forth. Thus the digit di is a
1 unless the ith digit of the ith number is a 1 in which case di = 2. Can you see that the number d
constructed in this manner is not on the list?
This algorithm shows that in any list of real numbers between 0 and 1, there is always at least
one number missing. Therefore, there is no complete list of these numbers. Thus there are
infinite sets of different sizes and the set of real numbers has a cardinality larger than that of the
”
set of integers.
We generally use the symbol !0, read Aleph naught, using the first letter in the Hebrew
alphabet with the subscript 0, to represent the cardinality of countably infinite sets. The
cardinality of the real numbers is sometimes referred to as c for continuum and sometimes
referred to as !1.
2
See An Introduction to the History of Mathematics, 6th ed. by Howard Eves;
Saunders College Publishing: New York. pg 569.
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Writing, discussion, comprehension and exploration (WDCE)exercises:
1. Explain in your own words what infinite means.
2. Explain why the number d constructed in the algorithm used to prove theorem A.2 is not in
the list.
3. Research the Axiom of Choice. What is it? Is it or was it ever controversial?
Problems:
1. (Challenge) Prove that the two definitions of an infinite set are equivalent (i.e. that each
implies the other) point out where the Axiom of Choice comes in.
2. Construct an alternate algorithm for the proof of theorem A.2.
3. Show that if S and T are denumerable sets then ScT is denumerable. (hint: consider the
argument that the integers are countable.)
4. State an alternative definition of one-to-one correspondence using one-to-one and onto
functions.
Part II: Infinitely Many Infinities and the Continuum Hypothesis.
(This section of the appendix is best read after reading through Section 2.1.)
So we know there are at least two infinities, but are there more? How many? It turns out
that there are infinitely many different sizes of infinity. recall, from section 2.1 that if a set has n
elements then it has 2n subsets.
Definition A.7: (a.k.a.Definition 1.0.8) The set of all subsets of a set A is called the
Power Set of A and is denoted P (A).
What about the size of the power set of an infinite set, A. Can it be put in one-to-one
correspondence with the set itself? It turns out that the answer to this is no. Since there is an
obvious one-to-one correspondence between the elements of A and the single-element subsets in
P (A), then this means that the power set is bigger than the original set.
Theorem A.3: There is no one-to-one correspondence between a set and its power set.
The proof of this theorem is reminiscent of the famous paradox of Bertrand Russell (1872-1970)
regarding the set of all sets that do not contain themselves. It goes as follows:
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Proof:
The statement is clearly true for the empty set as it has no elements but its power set has one
element. Now suppose there is a one-to one correspondence between a non-empty set A and its
power set P (A). Then given an element x there is a subset of A to which x corresponds, we
will call this subset p(x). Now make the following definition: if x is in A then x is normal if
x0p(x) and otherwise it is not normal. Clearly each element in A is either normal or not normal,
thus one of the subsets of A contains all the elements of A that are not normal. Let us denote
this subset N. So there must be an element of A, call it n, which corresponds to this set.
Furthermore, n is either normal or it is not normal. Which is it? If it is not normal then it is in
N, in which case it is normal, which is a contradiction. Therefore, as the original assumption
leads to a contradiction, it is not possible to put a set in one-to-one correspondence with its
power set.
”
Now we know that there are denumerably many infinities, at least. Given the cardinality
of the counting numbers, !0, then define !1 to be the cardinality of P (N). !2 to be the
cardinality of P (P (N)), etc. (You may have guessed at this point that it is possible to show
that the set of real numbers and the set of subsets of the natural numbers are the same size.) In
this way we can construct an infinite sequence of increasing cardinalities. These infinite
cardinalities are also know as transfinite numbers. And there is a branch of mathematics which
studies transfinite numbers.
Now we know that there are a countably infinite number of transfinite numbers, but are
there more? In particular, are there any sets whose cardinality lies between !0 and !1 or, more
generally, between !i and !i+1? The famous Continuum Hypothesis, conjectures not. In about
19403, the Austrian logician, Kurt Gödel, famous for his incompleteness theorem, proved that the
continuum hypothesis is consistent with the accepted axioms of set theory, if they are consistent
with each other. In other words, he showed that it would be impossible to disprove the
continuum hypothesis in the standard realm of modern set theory. He also conjectured that it
would be impossible to prove it as well. In 1963, Paul J. Cohen of Stanford University proved
he was right. Thus the Continuum Hypothesis is undecidable on the basis of the axioms of set
theory. It is to the axioms of set theory, like the parallel postulate is to the rest of the axioms of
Geometry.
Writing, discussion, comprehension and exploration (WDCE)exercises:
1. Investigate Russell's paradox either in the library or on the internet. What can you find out
about it? Explain how it relates to Russell's story about a male barber who shaves all the men
and only the men who do not shave themselves. Who shaves the barber?
3
See An Introduction to the History of Mathematics, 6th ed. by Howard Eves;
Saunders College Publishing: New York. pg 617.
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2. A consequence of theorem A.3 is that there is no largest infinite set. Cantor, however, was
bothered by this consequence; what about the set of all sets? Can you reslove this paradox?
Explain.
3. The result that the Continuum Hypothesis is undecidable on the basis of the axioms of set
theory may make you a little uncomfortable. You may feel that mathematicians have been
remiss in developing a theory with insufficient axioms to answer all its questions. In their
defense I offer Gödel's Incompleteness Theorem in which Kurt Gödel proved that no set of
axioms could ever be sufficient to determine the answers to all questions. What can you find out
about this theorem and people's responses to it? What do you think about this result?
4. Can you find or construct a proof that the set of real numbers and the power set for the natural
numbers are the same size?
5. What do you know about the parallel postulate? How is it similar to the Continuum
Hypothesis?
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