Axiom Of Choice

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 73 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 \(\Rightarrow\) 17	The independence of Ramsey's theorem, Kleinberg, E.M. 1969, J. Symbolic Logic
17 \(\Rightarrow\) 132	Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
132 \(\Rightarrow\) 73	clear

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.
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.
132:	\(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function.
73:	\(\forall n\in\omega\), \(PC(\infty,n,\infty)\): For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b]

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