We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 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 |
| 52 \(\Rightarrow\) 142 |
The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics |
| 142 \(\Rightarrow\) 280 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 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 \((\forall x \in V)(f^{*}(x) \le p(x))\). |
| 142: | \(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7. |
| 280: | There is a complete separable metric space with a subset which does not have the Baire property. |
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