We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 213 \(\Rightarrow\) 85 | clear |
| 85 \(\Rightarrow\) 62 | clear |
| 62 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 11 | clear |
| 11 \(\Rightarrow\) 12 | clear |
| 12 \(\Rightarrow\) 73 |
Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc. Set Theory: Techniques and Applications, De la Cruz/Di Prisco, 1998b, 47-70 |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 213: | \(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function. |
| 85: | \(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13. |
| 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] |
| 73: | \(\forall n\in\omega\), \(PC(\infty,n,\infty)\): For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b] |
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