Implications

We have the following indirect implication of form equivalence classes:

168 \(\Rightarrow\) 167
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 \(\Rightarrow\) 376 clear
376 \(\Rightarrow\) 167 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.

376:

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

167:

\(PKW(\aleph_{0},\ge 2,\infty)\), Partial Kinna-Wagner Principle:  For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\).

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