Axiom Of Choice

We have the following indirect implication of form equivalence classes:

67 \(\Rightarrow\) 93 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
67 \(\Rightarrow\) 52	Independence of the prime ideal theorem from the Hahn Banach theorem, Pincus, D. 1972b, Bull. Amer. Math. Soc.
52 \(\Rightarrow\) 93	The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, Foreman, M. 1991, Fund. Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
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 \((\forall x \in V)(f^{*}(x) \le p(x))\).
93:	There is a non-measurable subset of \({\Bbb R}\).

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