Implications

This non-implication, Form 387 \( \not \Rightarrow \) Form 147, 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: 9681, whose string of implications is:
    82 \(\Rightarrow\) 387
  • A proven non-implication whose code is 3. In this case, it's Code 3: 145, Form 82 \( \not \Rightarrow \) Form 79 whose summary information is:
    Hypothesis Statement
    Form 82 <p> \(E(I,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) </p>

    Conclusion Statement
    Form 79 <p> \({\Bbb R}\) can be well ordered.  <a href="/articles/hilbert-1900">Hilbert [1900]</a>, p 263. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6831, whose string of implications is:
    147 \(\Rightarrow\) 91 \(\Rightarrow\) 79

The conclusion Form 387 \( \not \Rightarrow \) Form 147 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them
\(\cal M2(\langle\omega_2\rangle)\) Feferman/Truss Model This is another extension of <a href="/models/Feferman-1">\(\cal M2\)</a>
\(\cal M5(\aleph)\) 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 \(\Bbb R\)
\(\cal M6\) Sageev's Model I Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)
\(\cal M18\) 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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