Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 \(\Rightarrow\) 196-alpha given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
101 \(\Rightarrow\) 40	On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
40 \(\Rightarrow\) 86-alpha	clear
86-alpha \(\Rightarrow\) 196-alpha	Successive large cardinals, Bull Jr., E. L. 1978, Ann. Math. Logic

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

Howard-Rubin Number	Statement
101:	Partition Principle: If \(S\) is a partition of \(M\), then \(S \precsim M\).
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
86-alpha:	\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(\|X\| = \aleph_{\alpha }\), then \(X\) has a choice function.
196-alpha:	\(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable.

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