Axiom Of Choice

We have the following indirect implication of form equivalence classes:

41 \(\Rightarrow\) 83 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
41 \(\Rightarrow\) 9	clear
9 \(\Rightarrow\) 83	The Axiom of Choice, Jech, 1973b, page 52 problem 9

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

Howard-Rubin Number	Statement
41:	\(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \).
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
83:	\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

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