Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 330
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
133 \(\Rightarrow\) 231 note-123
231 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 330 clear

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

Howard-Rubin Number Statement
133:  

Every set is either well orderable or has an infinite amorphous subset.

231:

\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.

165:

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

330:

\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See Form 67.)

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