Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
129 \(\Rightarrow\) 4 clear
4 \(\Rightarrow\) 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
129:

For every infinite set \(A\), \(A\) admits a partition into sets of order type \(\omega^{*} + \omega\). (For every infinite \(A\), there is a set \(\{\langle C_j,<_j \rangle: j\in J\}\) such that \(\{C_j: j\in J\}\) is a partition of \(A\) and for each \(j\in J\), \(<_j\) is an ordering of \(C_j\) of type \(\omega^* + \omega\).)

4:

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

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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