Axiom Of Choice

We have the following indirect implication of form equivalence classes:

337 \(\Rightarrow\) 13 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
337 \(\Rightarrow\) 92	clear
92 \(\Rightarrow\) 94	clear
94 \(\Rightarrow\) 13	The Axiom of Choice, Jech, 1973b, page 148 problem 10.1

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
337:	\(C(WO\), uniformly linearly ordered): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).
92:	\(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.
94:	\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.
13:	Every Dedekind finite subset of \({\Bbb R}\) is finite.

Comment:

Back