Implications

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 19
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
345 \(\Rightarrow\) 43 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
43 \(\Rightarrow\) 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
345:

Rasiowa-Sikorski Axiom:  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

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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