Axiom Of Choice

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 325 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
2 \(\Rightarrow\) 3	On successors in cardinal arithmetic, Truss, J. K. 1973c, Fund. Math.
3 \(\Rightarrow\) 9	Cardinal addition and the axiom of choice, Howard, P. 1974, Bull. Amer. Math. Soc.
9 \(\Rightarrow\) 325	note-46

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

Howard-Rubin Number	Statement
2:	Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).
3:	\(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
325:	Ramsey's Theorem II: \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see Form 17.)

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