Axiom Of Choice

We have the following indirect implication of form equivalence classes:

23 \(\Rightarrow\) 23 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
23 \(\Rightarrow\) 23	Zermelo's Axiom of Choice, Moore, [1982]

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

Howard-Rubin Number	Statement
23:	\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(\|\bigcup A\| = \aleph _{\alpha }\).
23:	\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(\|\bigcup A\| = \aleph _{\alpha }\).

Comment:

Back