We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
15 \(\Rightarrow\) 30 | The Axiom of Choice, Jech, 1973b, page 53 problem 4.12 |
30 \(\Rightarrow\) 62 | clear |
62 \(\Rightarrow\) 178-n-N | clear |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
15: | \(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)). |
30: | Ordering Principle: Every set can be linearly ordered. |
62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
178-n-N: | If \(n\in\omega\), \(n\ge 2\) and \(N \subseteq \{ 1, 2, \ldots , n-1 \}\), \(N \neq\emptyset\), \(MC(\infty,n, N)\): If \(X\) is any set of \(n\)-element sets then there is a function \(f\) with domain \(X\) such that for all \(A\in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\). |
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