We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 28-p \(\Rightarrow\) 427 | clear |
| 427 \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 52 |
Independence of the prime ideal theorem from the Hahn Banach theorem, Pincus, D. 1972b, Bull. Amer. Math. Soc. |
| 52 \(\Rightarrow\) 221 | clear |
| 221 \(\Rightarrow\) 222 | clear |
| 222 \(\Rightarrow\) 223 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 28-p: | (Where \(p\) is a prime) AL20(\(\mathbb Z_p\)): Every vector space \(V\) over \(\mathbb Z_p\) has the property that every linearly independent subset can be extended to a basis. (\(\mathbb Z_p\) is the \(p\) element field.) Rubin, H./Rubin, J. [1985], p. 119, Statement AL20 |
| 427: | \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis. \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}. |
| 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 \((\forall x \in V)(f^{*}(x) \le p(x))\). |
| 221: | For all infinite \(X\), there is a non-principal measure on \(\cal P(X)\). |
| 222: | There is a non-principal measure on \(\cal P(\omega)\). |
| 223: | There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\). |
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