Implications

We have the following indirect implication of form equivalence classes:

175 \(\Rightarrow\) 199(\(n\))
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
175 \(\Rightarrow\) 13 Horrors of topology without AC: A non-normal orderable space, van Douwen, E.K. 1985, Proc. Amer. Math. Soc.
note-49
13 \(\Rightarrow\) 199(\(n\)) clear

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

Howard-Rubin Number Statement
175:

Transitivity Condition: For all sets \(x\), there is a set \(u\) and a function \(f\) such that \(u\) is transitive and \(f\) is a one to one function from \(x\) onto \(u\).

13:

Every Dedekind finite subset of \({\Bbb R}\) is finite.

199(\(n\)):

(For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite.

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