Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 \(\Rightarrow\) 111 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 \(\Rightarrow\) 122	clear
122 \(\Rightarrow\) 250	clear
250 \(\Rightarrow\) 111	clear

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.
122:	\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.
250:	\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.
111:	\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

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