Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
287 \(\Rightarrow\) 222
222 \(\Rightarrow\) 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
287:

The Hahn-Banach Theorem for Separable Normed Linear Spaces:  Assume \(V\) is a separable normed linear space and \(p :V \to \Bbb R\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t \ge 0)(\forall x \in V)(p(tx) = tp(x))\) and assume \(f\) is a linear function from a subspace \(S\) of \(V\) into \(\Bbb R\) which satisfies \((\forall x \in S)(f(x) \le p(x))\), then \(f\) can be extended to \(f^* :V \to \Bbb R\) so that \(f^* \) is linear and \((\forall x \in V)(f^*(x) \le p(x))\).

222:

There is a non-principal measure on \(\cal P(\omega)\).

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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