Axiom Of Choice

This non-implication, Form 423 \( \not \Rightarrow \) 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: 10054, whose string of implications is: 10 \(\Rightarrow\) 423

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 423 \( \not \Rightarrow \) 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>