Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
394 \(\Rightarrow\) 337 clear
337 \(\Rightarrow\) 92 clear
92 \(\Rightarrow\) 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
394:

\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.

337:

\(C(WO\), uniformly linearly ordered):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).

92:

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

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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