Implications

We have the following indirect implication of form equivalence classes:

168 \(\Rightarrow\) 336-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
168 \(\Rightarrow\) 100 clear
100 \(\Rightarrow\) 9 On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc.
The Axiom of Choice, Jech, 1973b, page 162 problem 11.8
9 \(\Rightarrow\) 336-n clear

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

Howard-Rubin Number Statement
168:

Dual Cantor-Bernstein Theorem:\((\forall x) (\forall y)(|x| \le^*|y|\) and \(|y|\le^* |x|\) implies  \(|x| = |y|)\) .

100:

Weak Partition Principle:  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

336-n:

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.

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