Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
130 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 13 The Axiom of Choice, Jech, 1973b, page 148 problem 10.1
13 \(\Rightarrow\) 199(\(n\)) clear

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

Howard-Rubin Number Statement
130:

\({\cal P}(\Bbb R)\) is well orderable.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

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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