Implications

We have the following indirect implication of form equivalence classes:

214 \(\Rightarrow\) 405
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
214 \(\Rightarrow\) 152 note-140
152 \(\Rightarrow\) 4 Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math.
note-27
note-27
note-27
4 \(\Rightarrow\) 405 clear

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

Howard-Rubin Number Statement
214:

\(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\).

152:

\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.  (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)

4:

Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)

405:

Every infinite set can be partitioned into sets each of which is countable and has at least two elements.

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