502 – Equivalents of the axiom of choice

November 11, 2009

The goal of this note is to show the following result:

Theorem 1 The following statements are equivalent in {{\sf ZF}:}

 

  1. The axiom of choice: Every set can be well-ordered.
  2. Every collection of nonempty set admits a choice function, i.e., if {x\ne\emptyset} for all {x\in I,} then there is {f:I\rightarrow\bigcup I} such that {f(x)\in x} for all {x\in I.}
  3. Zorn’s lemma: If {(P,\le)} is a partially ordered set with the property that every chain has an upper bound, then {P} has maximal elements.
  4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set {S} such that {|S\cap x|=1} for all {x} in the family.
  5. Any set is a well-ordered union of finite sets of bounded size, i.e., for every set {x} there is a natural {m,} an ordinal {\alpha,} and a function {f:\alpha\rightarrow{\mathcal P}(x)} such that {|f(\beta)|\le m} for all {\beta<\alpha,} and {\bigcup_{\beta<\alpha}f(\beta)=x.}
  6. Tychonoff’s theorem: The topological product of compact spaces is compact.
  7. Every vector space (over any field) admits a basis.

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580 -Some choiceless results (2)

January 25, 2009

There are a few additional remarks on the Schröder-Bernstein theorem worth mentioning. I will expand on some of them later, in the context of descriptive set theory.

The dual Schröder-Bernstein theorem (dual S-B) is the statement “Whenever A,B are sets and there are surjections from A onto B and from B onto A, then there is a bijection between A and B.”

* This follows from the axiom of choice. In fact, {\sf AC} is equivalent to: Any surjective function admits a right inverse. So the dual S-B follows from choice and the S-B theorem. 

* The proofs of S-B actually show that if one has injections f:A\to B and g:B\to A, then one has a bijection h:A\to B contained in f\cup g^{-1}. So the argument above gives the same strengthened version of the dual S-B. Actually, over {\sf ZF}, this strengthened version implies choice. This is in Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375-381. 

* If j : {}x \to y is onto, then there is k:{\mathcal P}(y)\to {\mathcal P}(x) 1-1, so if there are surjections in both directions between A and B, then {\mathcal P}(A) and {\mathcal P}(B) have the same size. Of course, this is possible even if A and B do not.

Open question. ({\sf ZF}) Does the dual Schröder-Bernstein theorem imply the axiom of choice?

* The dual S-B is not a theorem of {\sf ZF}.

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