Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 225, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9853, whose string of implications is:
    52 \(\Rightarrow\) 309
  • A proven non-implication whose code is 3. In this case, it's Code 3: 230, Form 52 \( \not \Rightarrow \) Form 206 whose summary information is:
    Hypothesis Statement
    Form 52 <p> <strong>Hahn-Banach Theorem:</strong>  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 \((\forall x \in V)(f^{*}(x) \le p(x))\). </p>

    Conclusion Statement
    Form 206 <p> The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5889, whose string of implications is:
    225 \(\Rightarrow\) 70 \(\Rightarrow\) 206

The conclusion Form 309 \( \not \Rightarrow \) Form 225 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M27\) Pincus/Solovay Model I Let \(\cal M_1\) be a model of \(ZFC + V =L\)

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