- Oct 2024
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Cohen (Independence of the Axiomof Choice; The Independence of the Continuum Hypothesis I, I1) completed theproof of independence for each by showing neither could be deduced from theexisting axioms (by showing the negation of each could consistently be added tothe Zermelo—Fraenkel axiom scheme). See P. J. Cohen (Set Theory and theContinuum Hypothesis) for a discussion of these results and his intuition about thecontinuum hypothesis. Another expository reference is Cohen (IndependenceResults in Set Theory).
In 1963 Paul Cohen completed the work of Gödel by proving the independence of the axiom of choice and the continuum hypothesis from the Zermelo-Fraenkel set theory axioms. He did this by showing that neither could be deduced from the existing axioms and specifically by showing that the negation of each could be added to ZF consistently.
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Godel (The Consistency of the Axiom of Choice and of the Generalized Con-tinuum Hypothesis with the Axioms of Set Theory) proved in 1940 that additionof either the axiom of choice or the continuum hypothesis to existing set theoreticaxioms would not produce a contradiction.
Gödel's breakthrough in 1940 was to prove that one could extend the axioms of set theory to include the axiom of choice or the continuum hypothesis without introducing contradictions.
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- Sep 2024
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Z orn ’s lemma. Suppose S, < is a partially ordered set with the property that every chain in S has an upper bound. Then S contains amaximal element.
typo : < should be ≤
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The axiom o f choice. Suppose {Si}, i E I, is a family of nonemptysets. Then there is a function / from I into U / Si such that f(i) E Sifor each i e I.
For any collection of non-empty sets, one can create a set by choosing one element from each set in the given collection.
There are a variety of other equivalent ways to state this as well as names. One variation is Zorn's lemma.
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