We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 400 \(\Rightarrow\) 401 | clear |
| 401 \(\Rightarrow\) 327 | clear |
| 327 \(\Rightarrow\) 250 | clear |
| 250 \(\Rightarrow\) 47-n | clear |
| 47-n \(\Rightarrow\) 288-n | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 400: | \(KW(LO,LO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). |
| 401: | \(KW(LO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). |
| 327: | \(KW(WO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite 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.) |
| 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. |
| 47-n: | If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. |
| 288-n: | If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. |
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