Hypothesis: HR 222:
There is a non-principal measure on \(\cal P(\omega)\).
Conclusion: HR 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 \((\forall x \in V)(f^{*}(x) \le p(x))\).
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M2(\kappa)\) Feferman/Pincus Model | This is an extension of <a href="/models/Feferman-1">\(\cal M2\)</a> in which there are \(\kappa\) generic sets, where \(\kappa\) is a regular cardinal |
| \(\cal N51\) Weglorz/Brunner Model | Let \(A\) be denumerable and \(\cal G\)be the group of all permutations of \(A\) |
Code: 3
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