Implications

We have the following indirect implication of form equivalence classes:

333 \(\Rightarrow\) 413
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
333 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 144 Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 413 Constructive order theory, Ern'e, M. 2001, Math. Logic Quart.

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

Howard-Rubin Number Statement
333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

144:

Every set is almost well orderable.

413:

Every infinite set \(S\) is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to \(S\).  

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