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 \(\Rightarrow\) 285 |
On functions without fixed points, Wi'sniewski, K. 1973, Comment. Math. Prace Mat. |
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. |
285: | Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\). |
Comment: