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 \(\Rightarrow\) 82 | clear |
82 \(\Rightarrow\) 387 |
"Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math. |
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. |
82: | \(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) |
387: | DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)). |
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