Implications

We have the following indirect implication of form equivalence classes:

4 \(\Rightarrow\) 13
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
4 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 13 clear

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

Howard-Rubin Number Statement
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.

13:

Every Dedekind finite subset of \({\Bbb R}\) is finite.

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