We have the following indirect implication of form equivalence classes:
| 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 \(\Rightarrow\) 64 |
The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear |
| 64 \(\Rightarrow\) 127 |
Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung |
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. |
| 64: | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 127: | An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable. |
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