Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
212 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 34 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
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
212:

\(C(2^{\aleph_{0}},\subseteq{\Bbb R})\): If \(R\) is a relation on \({\Bbb R}\) such that for all \(x\in{\Bbb R}\), there is a \(y\in{\Bbb R}\) such that \(x\mathrel R y\), then there is a function \(f: {\Bbb R} \rightarrow{\Bbb R}\) such that for all \(x\in{\Bbb R}\), \(x\mathrel R f(x)\).

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

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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