This non-implication,
Form 0 \( \not \Rightarrow \)
Form 164,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 0 | \(0 = 0\). |
| Conclusion | Statement |
|---|---|
| Form 79 | <p> \({\Bbb R}\) can be well ordered. <a href="/articles/hilbert-1900">Hilbert [1900]</a>, p 263. </p> |
The conclusion Form 0 \( \not \Rightarrow \) Form 164 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 \(ZF + V = L\) |
| \(\cal M7\) Cohen's Second Model | There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model |
| \(\cal M9\) Feferman/Levy Model | Assume the ground model, \(\cal M\), satisfies \(ZF + GCH\) (the <strong>Generalized Continuum Hypothesis</strong>) |
| \(\cal M12(\aleph)\) Truss' Model I | This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular |
| \(\cal M15\) Feferman/Blass Model | Blass constructs a model similar to Feferman's model, <a href="/models/Feferman-1">\(\cal M2\)</a> |
| \(\cal M17\) Gitik's Model | Using the assumption that for every ordinal \(\alpha\) there is a strongly compact cardinal \(\kappa\) such that \(\kappa >\alpha\), Gitik extends the universe \(V\) by a filter \(G\) generic over a proper class of forcing conditions |
| \(\cal M18\) 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) |
| \(\cal M26\) Kanovei's Model I | Starting with a model of \(ZF + V = L\) and using forcing techniques due to <a href="/excerpts/Jensen-1968">Jensen [1968]</a>, Kanovei constructs a model of \(ZF\) in which there is an infinite Dedekind finite set \(A\) of generic reals that is in the class \(\varPi^1_n\), but there are no infinite Dedekind finite subsets of \(\Bbb R\) in the class \(\varSigma^1_n\), where \(n\in\omega\), \(n\ge 2\) |
| \(\cal M27\) 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) |
| \(\cal M36\) Figura's Model | Starting with a countable, standard model, \(\cal M\), of \(ZFC + 2^{\aleph_0}=\aleph_{\omega +1}\), Figura uses forcing conditions that are functions from a subset of \(\omega\times\omega\) to \(\omega_\omega\) to construct a symmetric extension of \(\cal M\) in which there is an uncountable well ordered subset of the reals (<a href="/form-classes/howard-rubin-170">Form 170</a> is true), but \(\aleph_1= \aleph_{\omega}\) so \(\aleph_1\) is singular (<a href="/form-classes/howard-rubin-34">Form 34</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> |
| \(\cal M41\) Kanovei's Model III | Let \(\Bbb P\) be the set of conditions from the model in <a href="/excerpts/Jensen-1968">Jensen [1968]</a> |