Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 430-p, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 241, Form 0 \( \not \Rightarrow \) Form 52 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5114, whose string of implications is:
    430-p \(\Rightarrow\) 67 \(\Rightarrow\) 52

The conclusion Form 0 \( \not \Rightarrow \) Form 430-p then follows.

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

Name Statement
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them
\(\cal M2(\kappa)\) 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
\(\cal M5(\aleph)\) Solovay's Model An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)
\(\cal M18\) Shelah's Model I Shelah modified Solovay's model, <a href="/models/Solovay-1">\(\cal M5\)</a>, and constructed a model without using an inaccessible cardinal in which the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true and every set of reals has the property of Baire (<a href="/form-classes/howard-rubin-142">Form142</a> is false)
\(\cal M30\) Pincus/Solovay Model II In this construction, an \(\omega_1\) sequence of generic reals is added to a model of \(ZFC\) in such a way that the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true, but no nonprincipal measure exists (<a href="/form-classes/howard-rubin-223">Form 223</a> is false)
\(\cal M38\) Shelah's Model II In a model of \(ZFC +\) "\(\kappa\) is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse \(\kappa\) to \(\aleph_1\) similarly to <a href="/articles/Solovay-1970">Solovay [1970]</a>
\(\cal N51\) Weglorz/Brunner Model Let \(A\) be denumerable and \(\cal G\)be the group of all permutations of \(A\)

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