We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 345 \(\Rightarrow\) 14 |
Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal |
| 14 \(\Rightarrow\) 52 |
On the application of Tychonoff's theorem in mathematical proofs, L o's, J. 1951, Fund. Math. Two applications of the method of construction by ultrapowers to analysis, Luxemburg, W.A.J. 1970, Proc. Symp. Pure. Math. Applications of Model Theory to Algebra, Analysis and Probability, Luxemburg, 1969, 123-137 |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 345: | Rasiowa-Sikorski Axiom: If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). |
| 14: | BPI: Every Boolean algebra has a prime ideal. |
| 52: | Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\). |
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