We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
43 \(\Rightarrow\) 8 | clear |
8 \(\Rightarrow\) 379 | clear |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
43: | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
8: | \(C(\aleph_{0},\infty)\): |
379: | \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). |
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