Axiom Of Choice

We have the following indirect implication of form equivalence classes:

168 \(\Rightarrow\) 17 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 \(\Rightarrow\) 17	The independence of Ramsey's theorem, Kleinberg, E.M. 1969, J. Symbolic Logic

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.
17:	Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20.

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