Axiom Of Choice

We have the following indirect implication of form equivalence classes:

109 \(\Rightarrow\) 280 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
109 \(\Rightarrow\) 66	clear
66 \(\Rightarrow\) 67	Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
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
109:	Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.
66:	Every vector space over a field has a basis.
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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