Axiom Of Choice

We have the following indirect implication of form equivalence classes:

302 \(\Rightarrow\) 93 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
302 \(\Rightarrow\) 392	clear
392 \(\Rightarrow\) 394	clear
394 \(\Rightarrow\) 337	clear
337 \(\Rightarrow\) 92	clear
92 \(\Rightarrow\) 170	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
170 \(\Rightarrow\) 93	Zermelo's Axiom of Choice, Moore, [1982]

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

Howard-Rubin Number	Statement
302:	Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.
392:	\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.
394:	\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.
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.
170:	\(\aleph_{1}\le 2^{\aleph_{0}}\).
93:	There is a non-measurable subset of \({\Bbb R}\).

Comment:

Back