Implications

This non-implication, Form 288-n \( \not \Rightarrow \) Form 385, 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: 10286, whose string of implications is:
    10 \(\Rightarrow\) 288-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 95, Form 10 \( \not \Rightarrow \) Form 385 whose summary information is:
    Hypothesis Statement
    Form 10 <p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

    Conclusion Statement
    Form 385 <p> <strong>Countable Ultrafilter Theorem</strong>:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter. </p>

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

The conclusion Form 288-n \( \not \Rightarrow \) Form 385 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
\(\cal M5(\aleph)\) 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)
\(\cal M27\) Pincus/Solovay Model I Let \(\cal M_1\) be a model of \(ZFC + V =L\)
\(\cal M30\) 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>

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