Axiom Of Choice

We have the following indirect implication of form equivalence classes:

86-alpha \(\Rightarrow\) 80 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
86-alpha \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 80	clear

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

Howard-Rubin Number	Statement
86-alpha:	\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(\|X\| = \aleph_{\alpha }\), then \(X\) has a choice function.
8:	\(C(\aleph_{0},\infty)\):
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
80:	\(C(\aleph_{0},2)\): Every denumerable set of pairs has a choice function.

Comment:

Back