Hypothesis: HR 10:
\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
Conclusion: HR 385:
Countable Ultrafilter Theorem: Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.
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> |
Code: 3
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