Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
7 \(\Rightarrow\) 9 On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc.
The Axiom of Choice, Jech, 1973b, page 161 problem 11.6
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
7:

There is no infinite decreasing sequence of cardinals.

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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