Implications

We have the following indirect implication of form equivalence classes:

333 \(\Rightarrow\) 142
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
333 \(\Rightarrow\) 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

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

Howard-Rubin Number Statement
333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

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.

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