Implications

This non-implication, Form 142 \( \not \Rightarrow \) Form 206, 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: 10173, whose string of implications is:
    52 \(\Rightarrow\) 142
  • 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 142 \( \not \Rightarrow \) Form 206 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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