We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 399 \(\Rightarrow\) 323 | clear |
| 323 \(\Rightarrow\) 62 | note-70 |
| 62 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 46-K | clear |
| 46-K \(\Rightarrow\) 48-K | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 399: | \(KW(\infty,LO)\), The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). |
| 323: | \(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.) |
| 62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
| 61: | \((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function. |
| 46-K: | If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function. |
| 48-K: | If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\). |
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