Axiom of choice


From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931.


The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of sets is finite, or if a selection rule is available – some distinguishing property that happens to hold for exactly one element in each set.


The equivalence was conjectured by Schoenflies in 1905. Abstract algebra *Hahn embedding theorem: Every ordered abelian group G order-embeds as a subgroup of the additive group \mathbb{R}^\Omega endowed with a lexicographical order, where Ω is the set of Archimedean equivalence classes of Ω.


In 1906 Russell declared PP to be equivalent, but whether the partition principle implies AC is still the oldest open problem in set theory, and the equivalences of the other statements are similarly hard old open problems.


This equivalence was conjectured by Hahn in 1907. ==Stronger forms of the negation of AC== If we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.


The former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. ===Results requiring AC (or weaker forms) but weaker than it=== One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.


ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodin cardinals). Quine's system of axiomatic set theory, "New Foundations" (NF), takes its name from the title ("New Foundations for Mathematical Logic") of the 1937 article which introduced it.


Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned. ==Independence== In 1938, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) which satisfies ZFC and thus showing that ZFC is consistent if ZF itself is consistent.


In 1963, Paul Cohen employed the technique of forcing, developed for this purpose, to show that, assuming ZF is consistent, the axiom of choice itself is not a theorem of ZF.

North Holland, 1963.


Reissued by Elsevier, April 1970.


North Holland/Elsevier, July 1985, . George Tourlakis, Lectures in Logic and Set Theory.


PDF download via :Translated in: Jean van Heijenoort, 2002.


II: Set Theory, Cambridge University Press, 2003.


, available as a Dover Publications reprint, 2013, . Herman Rubin, Jean E.

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