Implications

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 194
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
113 \(\Rightarrow\) 8 Tychonoff's theorem implies AC, Kelley, J.L. 1950, Fund. Math.
Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
8 \(\Rightarrow\) 16 clear
16 \(\Rightarrow\) 194 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
113:

Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.

8:

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

16:

\(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function.

194:

\(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979].

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