We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 391 \(\Rightarrow\) 399 | clear |
| 399 \(\Rightarrow\) 323 | clear |
| 323 \(\Rightarrow\) 62 | note-70 |
| 62 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 11 | clear |
| 11 \(\Rightarrow\) 12 | clear |
| 12 \(\Rightarrow\) 336-n | clear |
| 336-n \(\Rightarrow\) 64 |
Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc. |
| 64 \(\Rightarrow\) 390 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 391: | \(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function. |
| 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. |
| 11: | A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b] |
| 12: | A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b] |
| 336-n: | (For \(n\in\omega\), \(n\ge 2\).) For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function. |
| 64: | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 390: | Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983]. |
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