Axiom Of Choice

We have the following indirect implication of form equivalence classes:

76 \(\Rightarrow\) 131 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
76 \(\Rightarrow\) 131	clear

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

Howard-Rubin Number	Statement
76:	\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).
131:	\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

Comment:

Back