Implications

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 250
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 40 clear
40 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 250 clear

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

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

250:

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

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