Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 \(\Rightarrow\) 361 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 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 361	Zermelo's Axiom of Choice, Moore, 1982, page 325

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.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
361:	In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.

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