Implications

We have the following indirect implication of form equivalence classes:

101 \(\Rightarrow\) 19
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 \(\Rightarrow\) 27 clear
27 \(\Rightarrow\) 31 clear
31 \(\Rightarrow\) 34 clear
34 \(\Rightarrow\) 19 Sur les fonctions representables analytiquement, Lebesgue, H. 1905, J. Math. Pures Appl.

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:

\(C(\aleph_{0},\infty)\):

27:

\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.

31:

\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

34:

\(\aleph_{1}\) is regular.

19:

A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1).

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