Implications

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 38
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 \(\Rightarrow\) 6 L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
6 \(\Rightarrow\) 5 L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
5 \(\Rightarrow\) 38 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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.

6:

\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable.

5:

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

38:

\({\Bbb R}\) is not the union of a countable family of countable sets.

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