Implications

We have the following indirect implication of form equivalence classes:

86-alpha \(\Rightarrow\) 128
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
86-alpha \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 128 Realisierung und Auswahlaxiom, Brunner, N. 1984f, Arch. Math. (Brno)

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
86-alpha:

\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.

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.

128:

Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points.

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