We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 430-p \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 52 |
Independence of the prime ideal theorem from the Hahn Banach theorem, Pincus, D. 1972b, Bull. Amer. Math. Soc. |
| 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 |
|---|---|
| 430-p: | (Where \(p\) is a prime) \(AL21\)\((p)\): Every vector space over \(\mathbb Z_p\) has the property that for every subspace \(S\) of \(V\), there is a subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\) generates \(V\) in other words such that \(V = S \oplus S'\). Rubin, H./Rubin, J [1985], p.119, AL21. |
| 67: | \(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). |
| 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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