Cardinal Numbers
To every set E, we can assign an object lEl called the cardinal number of E and this can
be done in such a way that lEl œ lFl iff there exists a bijection between the sets E and
F . We also say that the set E has cardinality lElÞ
Informally, lEl is a “number” (possibly a new “infinite number”) that states how many
elements are in the set E.”
(To prove that this can be done for every set requires a substantial amount of work in
axiomatic set theory. And when this work is carefully done, the “object”
lEl œ “the cardinal number of E” is itself some set  because “everything is a set” in
mathematics. For our purposes, we simply assume that this plausible statement is true.)
We already gave special names to come cardinal numbers:
lgl œ !ß ÞÞÞ ß lÖ"ß #ß ÞÞÞß 5×l œ 5 , ...ß ll œ i! and lÐ!ß "Ñl œ -Þ
Since (for example)  ¶ ™ ¶  ¶  ¶  ‚  ‚ ™ ‚  ¶  ‚ ,
all of these sets have cardinality i! Þ
Since (for example) Ð!ß "Ñ ¶ Ð  &ß "(Ñ ¶ Ò  &ß "(Ñ ¶ Ð$ß %Ó ¶ Ò!ß "Ó ¶ ‘ ß
all of these sets have cardinality - .
Comparing the size of cardinal numbers
Definition For two cardinal numbers lEl and lFl, we write
i) lEl Ÿ lFl if there is a one-to-one function 0 À E Ä FÞ Since this means that
E ¶ rangeÐ0 Ñ © F , we can also say that “lEl Ÿ lFl if E is equivalent to a subset
of F .”
ii) lEl  lFl if lEl Ÿ lFl but lEl Á lFl
Thusß El  lFl if E is equivalent to a subset of F but E is not equivalent to F .
Of course, looking at !ß "ß #ß ... and thinking of them as cardinal numbers of sets, this
definition for Ÿ and  coincides with the usual meaning of Ÿ and  for whole
numbers: for example, thought of as cardinal numbers, #  & because there is a one-toone function 0 À Ö"ß #× Ä Ö"ß #ß $ß %ß &× but Ö"ß #× ¶
Î Ö"ß #ß $ß %ß &×.
Example Is i!  - ?
Joe argues:
ll œ i! and lÐ!ß "Ñl œ -Þ Since there exists a one-to-one
"
function 0 À  Ä Ð!ß "Ñ (for example, 0 Ð8Ñ œ #8
works), we see that i! Ÿ - . Since we
already proved that  ¶
Î Ð!ß "Ñ, we get that i!  - .
Jane argues: ll œ i! and l‘l œ - . Since there exists a one-to-one
function 0 À  Ä ‘ (for example, 0 Ð;Ñ œ ; works), we see that i! Ÿ -Þ Since we
already know that  ¶
Î ‘, we conclude that i!  -Þ
Joe and Jane “calculated” differently but got the same answer; the reasons appears to be
that  ¶  and Ð!ß "Ñ ¶ ‘ so that it didn't matter which set with cardinality i! and
which set with cardinality - was used in the calculation. Will this always be true?
That's what the next theorem says: that our definitions for Ÿ and  between cardinal
numbers are well defined.
Theorem If E ¶ F and G ¶ Hß then lEl Ÿ lGl iff lFl Ÿ lHlÞ Equivalently, in terms
of functions:
Suppose there exist bijections 9 À E Ä F and < À G Ä H. Then
there is a one-to-one function 0 À E Ä G iff there is a one-to-one function 1 À F Ä H
Proof (The proof should be very easy if you understand the meaning of one-to-one,
inverse functions, and function composition.)
By hypothesis, there are two bijections 9 and < as shown in the diagram below. Suppose
also there is a one-to-one function 0 À E Ä G as pictured:
E
9
Ä
F
<
Ä
H
Æ0
G
Since 9 and < are bijections, each has an inverse function, and using 0 we can define a
new function 1 À F Ä H as follows
1Ð,Ñ œ Ð< ‰ 0 ‰ 9" ÑÐ,Ñ œ <Ð0 Ð9" Ð,ÑÑÑÞ
Because 9" ß 0 , and < are one-to-one, so is 1Þ
Conversely, if we are given that there exists a one-to-one function 1 À F Ä H, we can
prove there is a one-to-one function 0 À E Ä G by defining 0 œ <" ‰ 1 ‰ 9Þ ñ
Example There is a one-to-one function 0 À  Ä c ÐÑ (for example, 0 Ð8Ñ œ Ö8×
works), so i! Ÿ lc ÐÑlÞ And since we know that  ¶
Î c ÐÑ, we can actually say that
i!  lc ÐÑlÞ
In a completely similar way, lEl  lc ÐEÑl for every set E.
For any set E, Ö!ß "×E stands for the set of all functions with domain E and codomain
Ö!ß "×Þ Thus, if 0 − Ö!ß "×E and + − E, we have either 0 Ð+Ñ œ ! or 0 Ð+Ñ œ "Þ
Theorem We claim that Ö!ß "×E ¶ c ÐEÑ
Proof We need to define a bijection F À Ö!ß "×E Ä c ÐEÑ. In makinbg a definition for
F: if 0 − Ö!ß "×E , we will need that FÐ0 Ñ − c (A)ß that is, FÐ0 Ñ © E
If 0 − Ö!ß "×E , let F0 œ Ö+ − E À 0 Ð+Ñ œ "×Þ
F0 is a subset of E. Define FÐ0 Ñ œ F0 , so that F À Ö!ß "×E Ä c ÐEÑ.
F is one-to-one: Suppose 0 ß 1 − Ö!ß "×E and 0 Á 1. Then b+ − E such that
0 Ð+Ñ Á 1Ð+Ñ: one is !, the other is " À without loss of generality, suppose
1Ð+Ñ œ ! and 0 Ð+Ñ œ "Þ Then + − F0 but +  F1 , so FÐ0 Ñ Á FÐ1ÑÞ
F is onto: Suppose F − c ÐEÑ, that is, suppose F © EÞ Define a function 0 with
domain E by
0 Ð+Ñ œ œ
"
!
if + − F
if + − E  F
Note: 0 is called the characteristic function of the set F ; this function is sometimes
denoted by ;F (where ; is the Greek letter “chi”)
Then 0 − Ö!ß "×E and FÐ0 Ñ œ F0 œ Ö+ − E À 0 Ð+Ñ œ "× œ FÞ
So F − rangeÐFÑÞ ñ
Example c ÐÑ ¶ Ö!,1× Þ A function 0 − Ö!,1× is a sequence in the set Ö!ß "×  that
is, 0 is a “binary sequence.” In more informal sequence notation, 0 is a sequence like
Ð"ß !ß !ß "ß !ß "ß ÞÞÞÑ In other words c ÐÑ is equivalence to the “set of all binary
sequences.” A bit informally, you can think of the correspondence established in the
preceding theorem in the following way:
A binary sequence like Ð"ß !ß !ß "ß !ß "ß ÞÞÞÑ is thought of as instructions
(yes, no, no, yes, no, yes,...)
telling us which natural numbers to include in a subset of : "ÐyesÑ, #ÐnoÑ,
3Ðno)ß ÞÞÞ
So Ð"ß !ß !ß "ß !ß "ß ÞÞÞÑ defines a subset of  that begins as Ö"ß %ß 'ß ÞÞÞ×Þ
Different binary sequences produces different subsets of  (one-to-one) and every subset
of  can be generated by a certain binary sequence (onto).
Example (See text, p. 244)
Imagine a programming language in which there is a set of allowed symbols such as:
+ , - Ð... etc) E F G (... etc) ! " # $ % & ' ( ) *
 ‡ Î
s Ð Ñ (and probably some other “keyboard” symbols that are used in this
language)
Suppose, for example, the total number of symbols available for writing in this language
happens to be 75 (a “blank space” corresponding to hitting the space bar on the
keyboard, is counted as one of these symbols.
A “string” of length is is some combination made up of 8 of these symbols end-to-end:
for example (counting also the spaces)
a*2(3 4c^)
is a string of length 10.
There are 75 possible strings of length ".
For a string of length 2, there are 75 choice for the first character and then 75 choices for
the second character: so there are altogether 75# possible strings of length 2
In general, there are 758 possible strings of length 8.
Most of these strings, if fed to the computer, would be treated as total meaningless
garbage. But certain strings if they are put together in just the right way accfordingto the
rules of the particular programming larguage, are programs that do or compute something
when you execute them.
Let T8 œ Ö: À : is a string of length 8 and : is a program×Þ Then lT8 l Ÿ (&8 œ the total
number of length 8 string. Probably, lT8 l is much, much less that (&8 Þ But the point is
that T8 is a finite set.
The coding for every program consists of only finitely many symbols, 8, wher perhaps 8
is very large. (Even if the program contains an “infinite loop” are runs “forever”, the
prograde code itself contains only finitely many symbols.)
Therefore the set of all possible programs T in the given language is countable, because
T œ -_
8œ" T8 which is a countable union of countable (in fact, finite) sets.
For example, we might write a program that generates as many terms as we like of a
certain binary sequence  perhaps the sequence Ð!ß "ß !ß !ß "ß !ß !ß !ß "ß ÞÞÞ, ).
But the set of all binary sequences œ Ö!ß "× is uncountable (it's equivalent to c ()).
Therefore the statement
For every binary sequence 0 − Ö!ß "× , these exists a program that generates 0
is FALSE. There just aren't enough possible programs.