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Some Set Theory We Should Know Cardinality and Cardinal Numbers

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Some Set Theory We Should Know Cardinality and Cardinal Numbers SOME SET THEORY WE SHOULD KNOW CARDINALITY AND CARDINAL NUMBERS De¯nition. Two sets A and B are said to have the same cardinality, and we write jAj = jBj, if there exists a one-to-one onto function f : A ! B. We also say jAj · jBj if there exists a one-to-one (but not necessarily onto) function f : A ! B. Then the SchrÄoder-BernsteinTheorem says: jAj · jBj and jBj · jAj implies jAj = jBj: SchrÄoder-BernsteinTheorem. If there are one-to-one maps f : A ! B and g : B ! A, then jAj = jBj. A set is called countable if it is either ¯nite or has the same cardinality as the set N of positive integers. Theorem ST1. (a) A countable union of countable sets is countable; (b) If A1;A2; :::; An are countable, so is ¦i·nAi; (c) If A is countable, so is the set of all ¯nite subsets of A, as well as the set of all ¯nite sequences of elements of A; (d) The set Q of all rational numbers is countable. Theorem ST2. The following sets have the same cardinality as the set R of real numbers: (a) The set P(N) of all subsets of the natural numbers N; (b) The set of all functions f : N ! f0; 1g; (c) The set of all in¯nite sequences of 0's and 1's; (d) The set of all in¯nite sequences of real numbers. The cardinality of N (and any countable in¯nite set) is denoted by @0. @1 denotes the next in¯nite cardinal, @2 the next, etc. Cantor's Continuum Hypothesis (CH) says that the cardinality of R is @1. It turns out that CH is undecidable from the usual axioms of set theory. The next theorem shows that for any cardinal, there is a bigger one. Theorem ST3. For any set X, we have jXj < jP(X)j, where P(X) is the set of all subsets of X. If the set X has cardinality ·, then the cardinal 2· is de¯ned to be the cardinality of the set P(X) (which has the same cardinality as the set of all functions from X into f0; 1g, by the same argument that gets ST2(a) () (b)). So by Theorem ST2, 2@0 is the cardinality of R, and Cantor's Continuum Hypothesis can be stated as @0 2 = @1. 1 2 SOME SET THEORY WE SHOULD KNOW ORDINAL NUMBERS AND TRANSFINITE INDUCTION De¯nition. A relation R on a set X is a subset of X £X. We write xRy to mean (x; y) 2 R.A partial order on X is a relation < satisfying, for every x; y; z 2 X: (i) x < y and y < x are not both true (antireflexive); (ii) x < y and y < z implies x < z (transitive). We also write x · y to mean \x < y or x = y". We say that < is a linear order if the following also holds for any x; y 2 X: (iii) Either x < y, y < x, or x = y (trichotomy). And ¯nally, a linear order < is a well-order if also: (iv) Every non-empty subset of X has a least element. Two linearly ordered sets (A; <A) and (B; <B) are order- isomorphic if there is a one-to-one onto order-preserving map h : A ! B. We will assume the following: Theorem ST4. There is a class Ord and a linear ordering < on Ord with the following properties: (i) Every nonempty subset of Ord has a least element; (ii) Given any well-ordered set (X; Á), there is ® 2 Ord such that (X; Á) is order-isomorphic to (P®; <), where P® = f¯ 2 Ord : ¯ < ®g. Elements of Ord are called ordinals. Intuitively, an ordinal number denotes position in a well-ordered sequence. The least ordinal is denoted by 0, the next by 1, the next by 2, etc. The least ordinal with in¯nitely many predecessors, i.e, the least ordinal greater than all the ¯nite ordinals 0; 1; 2;::: , is denoted by !. Then comes ! + 1;! + 2;::: . The least ordinal bigger than ! + n for every n < ! is denoted by ! ¢ 2. One can similarly de¯ne ! ¢ 3;! ¢ 4;::: . The least ordinal bigger than ! ¢ n for all n < ! is denoted by ! ¢ ! or !2. Etc. Note that all ordinals mentioned in the above paragraph have only countably many predecessors. Such ordinals are called countable ordinals. The least ordinal with uncountably many predecessors, i.e., the least ordinal greater than all of the countable ordinals, is denoted by !1. A better de¯nition of @1 than the one given above is that @1 denotes the cardinality of the set of predecessors of !1. @2; @3;::: may be de¯ned similarly. See below where @® is de¯ned, by trans¯nite induction, for any ordinal ®. Principle of Trans¯nite Induction. Let · be an ordinal. Suppose we have for each ® < · a statement S® such that: (i) S0 is true; (ii) If ¯ < · and S® is true for every ® < ¯, then S¯ is true. Then Sγ is true for every γ < ·. Theorem ST5. Let ® be any ordinal. Then the set [0; ®] = f¯ 2 On : 0 · ¯ · ®g with the order topology (using the usual ordering on the ordinals) is a compact Hausdor® space. Lemma ST6. Let ­ denote the set of countable ordinals. If C is any countable subset of ­, then there is a countable ordinal ® such that ® ¸ ¯ for any ¯ 2 C. SOME SET THEORY WE SHOULD KNOW 3 Theorem ST7. Give the set ­ of all countable ordinals the order topology (using the usual order on the ordinals). Then ­ with this topology is sequentially compact (and so also countably compact) but not compact. Principle of De¯nition by Trans¯nite Induction. Let · be an ordinal, B a set, and F the set of all functions (including the empty function) into B whose domain is the set f¯ 2 Ord : ¯ < ®g for some ® < ·. Let R : F! B. Then there is a unique function G with domain f® 2 Ord : ® < ·g such that, for any ® < ·, G(®) = R(G ¹ f¯ 2 Ord : ¯ < ®g) . Remark. The Trans¯nite Induction Principle and the Principle of De¯nition by Trans¯nite Induction also hold, by the same proof, if in their statements we replace · by the class Ord of all ordinals. Now the Principle of De¯nition by Trans¯nite Induction justi¯es de¯ning !® and @® for any ordinal ® as follows: (i) @0 is as de¯ned previously, and !0 = !. Suppose ® is an ordinal and !¯ and @¯ have been de¯ned for every ¯ < ®. (ii) If ® has an immediate predecessor γ (which would make ® = γ + 1), then let !® be the least ordinal with more than @γ -many predecessors, and let @® denote the cardinality of the set of predecessors of !®. (iii) If ® has no immediate predecessor, then let !® be the least ordinal greater than !¯ for every ¯ < ®. Again, let @® denote the cardinality of the set of predecessors of !®. Exercise ST8. (a) For each ordinal ® < !1, there is a one-to-one order-preserving function from f¯ 2 Ord : ¯ < ®g to the real line R; (b) There is no one-to-one order-preserving function from f¯ 2 Ord : ¯ < !1g to the real line R. AXIOM OF CHOICE AND EQUIVALENTS Axiom of Choice. Given any collection A of nonempty sets, there is a function f : A ! [A such that f(A) 2 A for every A 2 A. Zorn's Lemma. Suppose (P; <) is a partially ordered set with the following prop- erty: (¤) If L is any subset of P linearly ordered by <, then there is p 2 P such that l · p for any l 2 L. Then there is an element q in P such that no member of P is strictly greater than q. An element q of the partial order P that satis¯es the conclusion of Zorn's Lemma is called a maximal element of P . Theorem ST9. The following are equivalent: (a) The Axiom of Choice; (b) Zorn's Lemma; (c) Every set can be well-ordered..
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