We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
114 \(\Rightarrow\) 90 |
Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math. |
90 \(\Rightarrow\) 91 | The Axiom of Choice, Jech, 1973b, page 133 |
91 \(\Rightarrow\) 79 | clear |
79 \(\Rightarrow\) 92 | clear |
92 \(\Rightarrow\) 170 |
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 |
---|---|
114: | Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.) |
90: | \(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133. |
91: | \(PW\): The power set of a well ordered set can be well ordered. |
79: | \({\Bbb R}\) can be well ordered. Hilbert [1900], p 263. |
92: | \(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function. |
170: | \(\aleph_{1}\le 2^{\aleph_{0}}\). |
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