Axiom Of Choice

We have the following indirect implication of form equivalence classes:

101 \(\Rightarrow\) 151 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 \(\Rightarrow\) 231	Abzählbarkeit und Wohlordenbarkeit, Felgner, U. 1974, Comment. Math. Helv.
231 \(\Rightarrow\) 151	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.
231:	\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.
151:	\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)

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