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Cardinal
Numbers
- Introduction......................................................................................................................................
- Equipollence.....................................................................................................................................
- Finite and Infinite Sets .....................................................................................................................
- Denumerable Sets ............................................................................................................................
- Uncountable Sets..............................................................................................................................
- Infrapollence ....................................................................................................................................
- The Schroeder-Bernstein Theorem..................................................................................................
- Hyper-Uncountable Sets; The Continuum Hypothesis..................................................................
- The Cardinal Numbers ...................................................................................................................
1. Introduction
There are numerous mathematical generalizations of the natural numbers. On the one hand, there are the integers, the rational numbers, the real numbers, and the complex numbers. On the other hand, there are the cardinal numbers and the ordinal numbers. In the present chapter, we concentrate on cardinal numbers, leaving the other kinds of number to later chapters.
Natural numbers have both a cardinal use and an ordinal use. The former pertains to the quantitative notion of “how many”. The latter pertains to the order of the numbers rather than their sizes. In ordinary language syntax, the cardinal numerals are ‘1’, ‘2’, ‘3’, etc., whereas the ordinal numerals are ‘1st’, ‘2nd’, ‘3rd’, etc.
The cardinal (i.e., chief) use of numbers pertains to measuring how big sets are. This yields the derivative concept of the cardinality of a set A, which is how big A is, or how many elements A has. This suggests introducing a function sign ‘#(_)’, defined unofficially as follows.
(d) #(A) ¸ the number of elements in A
We need to provide a formal version of the definiens. Whatever we choose, the definition should have the following consequences. [Recall the numerical predicates from the previous chapter.]
(t0) #(A) = 0 ↔ 0[A] (t1) #(A) = 1 ↔ 1[A] (t2) #(A) = 2 ↔ 2[A] etc.
This just says the intuitively obvious – that the cardinality (size) of A is n iff A has n elements. One might be tempted to write the above series more succinctly as follows.
(t?) ∀n ( #(A) = n ↔ n[A] )
But this is ungrammatical; in first order logic, the same variable cannot appear in both singular term position and predicate position, as ‘n’ does in (t?). We have used the numerals ambiguously, as quantifiers, as predicates, and as singular terms; but only numerals-as-singular-terms can be quantificationally generalized in first order logic.
However, we can simulate (t?), by defining the expression ‘n[A]’, not to be the result of applying predicate variable ‘n’ to singular term ‘A’ (which cannot be accomplished in first order logic), but as applying a two-place predicate, written ‘[]’, to singular terms ‘n’ and ‘A’.
In the next section, we discuss how one can define ‘[]’ as a two-place predicate expression in set theory.
We are now in a position, finally, to define ‘n[A]’, at least in the finite case.
(d) n[A] ¸ n ∈ ω & n ¿ A
In other words, A has n elements iff there is a bijection from the number n onto A. Notice that, t 1 [t 2 ] is well-formed for any singular terms t 1 , t 2 , even if t 1 does not refer to a natural number. However, given the definition, t 1 [t 2 ] cannot be true unless t 1 denotes a natural number.
Notice the following natural consequences of our definitions.
(T4) ∀n(n ∈ ω ≤ n[n]) (T5) ∀n(n ∈ ω ≤ ∀yz(n[y] & n[z] .≤ y¿z))
In other words, every natural number n has n elements, and if A and B both have n elements (for any n), then A and B are equal in size.
We are now in a position to go back and define the function sign ‘#(_)’.
(d) #(A) = n ¸ n[A]
This has the following consequence, in conjunction with the earlier theorem.
(t) #(A) = #(B) ≤ A ¿ B
This says that if the number of A-elements is the same as the number of B-elements, then A and B are equipollent.
What we would like is the corresponding biconditional.
(??) #(A) = #(B) ↔ A ¿ B
But this is not true, at least the way we have defined ‘#()’ so far. The problem is that our definition of ‘#()’ applies only to finite sets.
Finitude and infinitude are the topics of the next section.
3. Finite and Infinite Sets
At this point, we are in a position, finally, to define the term ‘finite’ and the complementary term ‘infinite’.
(D2) fin[A]) ¸ ∃n(n ∈ ω & n[A])
In other words, A is finite iff it has n elements for some natural number n, which is to say that it is equipollent to some natural number.
To say that a set is infinite is simply to say that it is not finite:
(D3) infin[A]) ¸ ;fin[A]
Clearly, there are finite sets; for example, every natural number is finite, since every natural number is equipollent to a natural number – itself. Also, the singleton of any natural number is finite, having exactly one element.
The following are expected (if not easily proved) theorems about finite sets.
(T6) ∀x∀y ( fin[x] & y ⊆ x .≤ fin[y] ) (Any subset of a finite set is finite.)
(T7) ∀x ( fin[x] ≤ fin[”(x)] ) (The power set of any finite set is finite.)
(T8) ∀x∀y ( fin[x] & fin[y] .≤ fin[x∪y] ) (The union of two finite sets is finite.)
(T9) ∀x∀y ( fin[x] & fin[y] .≤ fin[x%y] ) (The Cartesian product of two finite sets is finite.)
(T10) ∀x ( fin[x] & ∀y(y ∈ x ≤ fin[y]) .≤ fin[Ÿx] ) (The union of a finite collection of finite sets is finite.)
The existence of finite sets is obvious. The existence of infinite sets is less obvious. However, the Axiom of Infinity (appropriately so called) does yield the existence of at least one infinite set. In particular, we have the following theorem.
(T11) ω is infinite
Proof: we proceed by induction, proving that no natural number is equipollent to ω. Base case: ω is not equipollent to 0; this is because only 0 is equipollent to 0, and ω is not 0. Inductive hypothesis: ω is not equipollent to m, to show: ω is not equipollent to m+. Suppose otherwise, to show a contradiction. Then there is a function that maps ω 1-1 onto m+, call it f. There is a unique element x in ω such that f(x)=m, call it b. Claim: f restricted to ω-{b} maps ω−{b} 1-1 onto m. Define function g from ω to ω−{b} as follows. For all x ‘ b, g(x)=x; for all x>b, f(x)=x+. Claim: g maps ω 1-1 onto ω−{b}. Claim: fôg maps ω 1-1 onto m, which means that ω is equipollent to m, which contradicts the inductive hypothesis.
4. Denumerable Sets
A set is finite iff it is equipollent to some element of ω; otherwise, it is infinite. On the other hand, a set is said to be denumerable (or denumerably infinite) if it is equipollent to the set ω of all natural numbers. Formally speaking,
(D4) den[A] ¸ A ¿ ω
Notice the following immediate theorem.
(T12) ∀X (den[X] ≤ infin[X] )
I.e., every denumerable set is infinite. Reason: if X is denumerable, it is equipollent to ω; if X is finite, it is equipollent to some natural number. So if X is both denumerable and finite, then ω is equipollent to some natural number, and hence is finite, which we have already proved is not true.
An alternative picture might be visually helpful.
(0,0)
(1,0) (0,1)
(2,0) (2,1) (2,2)
(3,0) (3,1) (3,2) (3,3)
(4,0) (4,1) (4,2) (4,3) (4,4)
… … … … … …
In this diagram, we simply count the first row, the second row, etc. It is evident that every ordered pair is on some row, and each row can be counted.
The same technique can be used to prove that the rational numbers are denumerable, which means that there are just as many natural numbers as there are rational numbers!
The above technique can also be used to prove the following.
(T14) If f is denumerable, and every element of f is denumerable, then Ÿf is denumerable. I.e., the union of any denumerable collection of denumerable sets is itself denumerable.
(T15) If f is a finite family of denumerable sets, then %f is denumerable. I.e., the Cartesian product of any finite family of denumerable sets is denumerable.
From what has transpired so far, one would naturally draw the conclusion that all infinite sets are denumerable, which implies in particular that all infinite sets have the same size. This is quite definitely not true, as we see in the next section.
5. Uncountable Sets
So far, we have discussed finite sets and denumerable sets, which are a species of infinite sets. It was suggested at the end of the last section that there are infinite sets that are not denumerable. Such sets are called uncountable (also, uncountably infinite, also nondenumerable ). The traditional usage is actually somewhat confusing, so let us chart the relationships.
(D2) A set is finite iff it is equipollent to some natural number. (D3) A set is infinite iff it is not finite. (D4) A set is denumerable iff it is equipollent to ω. (D5) A set is countable iff it is finite or denumerable. (D6) A set is uncountable (non-denumerable) iff it is infinite but not denumerable.
Notice in particular that there is overlap of the concepts; in particular, Infinite + Countable = Denumerable. Also notice the mildly illogical fact that ‘nondenumerable’ does not mean ‘not denumerable’. A set that is not denumerable (versus nondenumerable) may be either uncountable or finite, whereas a nondenumerable set cannot be finite.
Are there any uncountable sets? The affirmative answer was first proved by Cantor, who proved two examples, namely
(c1) the set of real numbers (rationals ∪ irrationals) is uncountable. (c2) the power set of ω is uncountable.
We have not discussed, nor even defined, the real numbers so far, so we must postpone the proof of (c1) until later. However, we already have the machinery to prove (c2).
(T16) ”(ω) is uncountable.
Proof: Let f be any function from ω into ”(ω). We wish to prove that f is not onto. For each number n, there is a unique subset f(n); n may or may not be an element of f(n). Call a number n normal if it is not an element of f(n). Consider the subset of normal numbers, call it N: N = {n : n ∉ f(n)}. Claim: N ∉ ran(f), and hence f is not onto ”(ω). Proof: Suppose N ∈ ran(f). Then N = f(m), where m ∈ ω. I.e., f(m) = {n : n ∉ f(n)}. It follows that m ∈ f(m) ↔ m ∉ f(m), which is a contradiction. This proves that ”(ω) is not denumerable; to prove that ”(ω) is uncountable (nondenumerable; recall distinction from above), we must further prove that ω is infinite. This is left as an exercise.
6. Infrapollence
The theorem that ”(ω) is uncountable is actually a special case of a more general theorem, which says that no set is equal in size to its power set, that every set is smaller in size than its power set. In order to state this theorem, we need to develop some additional notation and terminology.
Just as the identity of two sets, A and B, can be decomposed into two inclusions, A⊆B and B⊆A, the equipollence (equality of size) of two sets can be similarly decomposed. The relevant notion is what I propose to call infrapollence , which means “of lower (or equal) power ”. It is officially defined as follows.
(D7) A 9 B ¸ ∃f [ f:A≤B(1-1) ]
For example, consider {a}, {b}, where a≠b. Then {a} 9 {b} and {b} 9 {a}, but {a}≠{b}. On a more esoteric level, the set ω+ of positive natural numbers can be injected into ω, by the identity function, and ω can be injected into ω+, by the successor function. But ω≠ω+.
But what is the equivalence relation J? The answer is exactly what one might expect – equipollence. In other words, we have the following theorem, which is usually called the Schroeder- Bernstein Theorem
(T24) A 9 B & B 9 A .≤ A ¿ B
It is also called the Cantor-Schroeder-Bernstein Theorem; Cantor conjectured it, and Schroeder and Bernstein independently proved it in the 1890’s.
In order to prove The Schroeder-Bernstein Theorem, (T24), we first prove another important theorem, the Fixed Point Theorem (or rather, a special case).
Fixed Point Theorem (special case): (T25) Let A be any set, and let f be any function from ”(A) into ”(A) satisfying the following condition, called monotonicity.
(m) ∀XY[X ⊆ Y ≤ f(X) ⊆ f(Y)]
Then there is a "fixed point", i.e., a set Z such that f(Z)=Z.
Proof : Suppose f is such a function on ”(A). Consider the collection {X: X ⊆ f(X)}, call it f. Consider the union, Ÿf. We wish to show that Ÿf = f(Ÿf). Claim: ∀X[X ∈ f ≤ X ⊆ f(Ÿf)]. For suppose D ∈ f; then D ⊆ f(D); also D ⊆ Ÿf, so by (m), f(D) ⊆ f(Ÿf). So, D ⊆ f(Ÿf). So, by a previous theorem, Ÿf ⊆ f(Ÿf). We are half done. Now, Ÿf ⊆ f(Ÿf), so by (m), f(Ÿf) ⊆ f(f(Ÿf). But the latter means that f(Ÿf) ∈ f, from which we obtain the converse inclusion, f(Ÿf) ⊆ Ÿf. Thus, f(Ÿf) = Ÿf.
Next, we state a few relevant lemmas, whose proofs are left as exercises.
Lemma 1:
Let A and B be sets. Let A 1 and A 2 be disjoint subsets of A such that A 1 ∪A 2 = A. Similarly, let B 1 and B 2 be disjoint subsets of B such that B 1 ∪B 2 = B. Then:
A 1 ¿ B 1 & A 2 ¿ B 2 ≤. A ¿ B
Lemma 2:
Let A be any set, let X,Y be any subsets of A, and let f be any function on A. Then:
(1) X ⊆ Y ≤ A−Y ⊆ A−X (2) X ⊆ Y ≤ f≤(X) ⊆ f≤(Y)
Lemma 3 (a corollary to Lemma 2):
Let A,B be any sets, let f be any function from A to B, let g be any function from B into A, let X,Y be any subsets of A. Then:
X ⊆ Y ≤ g≤(B−f≤(A−X)) ⊆ g≤(B−f≤(A−Y))
Lemma 4: if f is a 1-1 function, and A ⊆ dom(f), then A ¿ f≤(A)
Having proven the fixed point theorem, and the relevant lemmas, we now prove
The Schroeder-Bernstein Theorem.
(T24) A 9 B & B 9 A .≤ A ¿ B
Proof: Suppose A 9 B & B 9 A, to show A ¿ B. Then there is an injection, f, from A into B. And there is an injection, g, from B into A. We use Lemma 1 to show A¿B; in particular, we divide A into disjoint parts A 1 , A 2 , and B into disjoint parts B 1 , B 2 , and we show the respective parts are equipollent. We divide A into parts by appealing to the Fixed Point Theorem. Specifically, by Lemma 3, the function that maps each subset X of A to the set g≤(B−f≤(A−X)) satisfies the requirements of (FTP), so there is fixed point, namely a set, call it K, such that g≤(B−f≤(A−K)) = K. Since A−K ⊆ dom(f), A−K ¿ f≤(A−K), by Lemma 4. Also, B−f≤(A−K) ⊆ dom(g), and K = g≤(B−f≤(A−K)), so K ¿ B−f≤(A−K), by Lemma 4. Furthermore, the sets in question divide A and B respectively, so applying Lemma 1, we have A ¿ B.
8. Hyper-Uncountable Sets; The Continuum Hypothesis
We proved earlier that the power set of ω is uncountable. In the present section, we use the same technique, in conjunction with the Schroeder-Bernstein theorem, to show that every set is strictly smaller than its power set. This is sometimes called Cantor’s Theorem.
(T25) A A ”(A)
Proof: Given the definition, we have to show A 9 ”(A), and we have to show ;[”(A) 9 A]. The former is shown by noting that the function f, defined so that f(x)={x} is an injection from A into ”(A). The latter is shown by showing ;[”(A) ¿ A], and appealing to the Schroeder-Bernstein Theorem. So suppose that ”(A) ¿ A. Then there is a 1-1 function from A onto ”(A). We prove that, in fact, there is no function from A onto ”(A). Once again, we construct the set of normal elements of A. N = {x: x ∉ f(x)}. Claim: N ∉ ran(f). For suppose otherwise. Then N = f(a), where a ∈ A. So f(a) = {x: x ∉ f(x), from which it follows that a ∈ f(a) ↔ a ∉ f(a), a contradiction.
Since every set is smaller than its power set, we have the following infinite chain.
A A ”(A) A ””(A) A ”””(A) A etc.
In the case that A is infinite, we obtain the result that there are infinitely many degrees of infinity. ”(ω) is bigger than ω, ””(ω) is bigger than ”(ω), ”””(ω) is bigger than ””(ω), etc. One might describe this by saying that ”(ω) is uncountable, ””(ω) is hyper-uncountable, ”””(ω) is hyper- hyper-uncountable, etc.
[What is worse, there is a set bigger than all of these – an infinitely-hyper-uncountable set! However, we do not prove that just yet.]
It appears that we have to do the same thing for the transfinite cardinals. For each “cardinality class”, we have to select a representative set of that cardinality. This includes picking a representative of each of the following.
(c1) an X such that X ¿ ω (c2) an X such that X ¿ ”(ω) (c3) an X such that X ¿ ””(ω) etc.
More generally, for each set A, we must select a representative set equipollent to A.
The official definition we adopt goes as follows.
(D9) #(A) ¸ the first ordinal number α such that α ¿ A
[Ordinals are discussed in detail in the next chapter.] Observe that this definition does not work properly unless we assume the Axiom of Choice. The ordinals are well-ordered by 9 , where a 9 b iff a∈b or a=b. So if there are any ordinals equipollent to a set A, there is a smallest such ordinal. The question, then, is whether every set is equipollent to at least one ordinal number. Here, we appeal to the Well-Ordering Theorem, which is a well-known equivalent of the Axiom of Choice. In particular, we well-order the set A (using AC), and then we appeal to a further theorem that every well-ordered set is isomorphic (and hence equipollent) to at least one ordinal.
For those who don’t accept the Axiom of Choice, other routes are available. One approach is to treat the expression ‘#(A)’ as syncategorimatic, as a pseudo-term, by which I mean that it only appears in formulas of the form ‘#(A)=#(B)’, and these formulas are defined as follows.
(d) #(A) = #(B) ¸ A ¿ B
A more common alternative approach is to introduce ‘#(_)’ as a further primitive item of set theory, and postulate the following further axiom.
(a1) ∀x∀y [ #(x)=#(y) ↔ x¿y ]
If this is all we postulate, then we cannot prove that the natural numbers, as defined earlier, are cardinal numbers. In this case, we have both the finite cardinals and the natural numbers. If this is undesirable, then we probably would add the following further postulates.
(a2) ∀x [ ∃n(n ∈ ω & x ¿ n .↔ #(x)=n ]
This has the desirable consequence that the finite cardinals are coextensive with the natural numbers. It does not say what the infinite cardinal numbers are, however. For example, the infinite cardinal numbers may be finite sets, for all we know. This is ok, so long as they are not natural numbers. The following might be true (for all we know).
#(ω) = {{∅}} #(”(ω)) = {{{∅}}} #(””(ω)) = {{{{∅}}}}
If we want the infinite cardinal numbers to be like the finite cardinal numbers, in the sense that they all have the “appropriate” size, then the following is a further plausible axiom.
(a3) ∀X[#(X) ¿ X]
This just says that the cardinal number of a set is the same size as the set. This rules out the above identifications.
Still another approach is to treat cardinal numbers as points (first elements), rather than as sets, in which case no cardinal number is identified with any set. This requires redoing the theory of natural numbers.
So long as we have the Fundamental Principle of Cardinal Numbers, either as a theorem deduced from the definition of cardinal number, or as an axiom within a theory that regards ‘#(_)’ as a primitive expression, we can deduce many properties of cardinal numbers.
For example, we can define addition, multiplication, and less-than-or-equal, as follows.
(D10) m+n=k ¸ ∃XY( X⊥Y & #(X)=m & #(Y)=n & #(X∪Y)=k ) (D11) mn=k ¸ ∃XY( #(X)=m & #(Y)=n & #(X%Y)=k ) (D12) m ≤ n ¸ ∃XY( #(X)=m & #(Y)=n & X 9 Y )
For example, one can prove that addition and multiplication are both associative and commutative, that multiplication distributes over addition, that ≤ is reflexive, transitive, and anti- symmetric. Curiously, one cannot prove that ≤ is connected; that is, one cannot prove
(C) m ≤ n ∨ m ≤ n,
without appealing to the Axiom of Choice; (C) is in fact equivalent to the Axiom of Choice. So, if one believes that infinite cardinal numbers are (or should be) arranged in a linear order just like the finite cardinal numbers, then one must further postulate (C), but this is tantamount to postulating the Axiom of Choice, in which case one might as well define the cardinal numbers to be first ordinal numbers, as originally suggested.
Mathematicians routinely employ the Axiom of Choice in their reasoning, because of its great power, although many use it apologetically. As shown by Cohen, AC is not a logical consequence of Zermelo-Fraenkel set theory, and hence is not an essential feature of the iterative conception of sets.
The latter is the idea that sets are generated from first elements by repeated (iterated) application of a few set-forming processes, including – pairs, unions, power sets, and replacement. These seem a necessary part of our conception of sets. By contrast, the epistemological status of the other axioms, including the Axiom of Choice, and the Continuum Hypothesis, remains unclear. This suggests the existence of alternative set theories on a par with alternative (non-Euclidean) geometries.
