We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
270 \(\Rightarrow\) 62 |
Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic |
62 \(\Rightarrow\) 121 | clear |
121 \(\Rightarrow\) 122 | clear |
122 \(\Rightarrow\) 327 | clear |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
270: | \(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas. |
62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
121: | \(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function. |
122: | \(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function. |
327: | \(KW(WO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite 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.) |
Comment: