Implications

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 199(\(n\))
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 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 13 clear
13 \(\Rightarrow\) 199(\(n\)) 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:

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

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

13:

Every Dedekind finite subset of \({\Bbb R}\) is finite.

199(\(n\)):

(For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite.

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