Implications

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 194
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
164 \(\Rightarrow\) 91 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 194 clear

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

Howard-Rubin Number Statement
164:

Every non-well-orderable set has an infinite subset with a Dedekind finite power set.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

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.

194:

\(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979].

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