Axiom Of Choice

This non-implication, Form 369 \( \not \Rightarrow \) Form 313, whose code is 4, is constructed around a proven non-implication as follows:

This non-implication was constructed without the use of this first code 2/1 implication.
A proven non-implication whose code is 3. In this case, it's Code 3: 1212, Form 369 \( \not \Rightarrow \) Form 280 whose summary information is:

Hypothesis Statement

Form 369 <p> If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). </p>

Conclusion Statement

Form 280 <p> There is a complete separable metric space with a subset which does not have the Baire property. </p>
An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7874, whose string of implications is: 313 \(\Rightarrow\) 222 \(\Rightarrow\) 142 \(\Rightarrow\) 280

Hypothesis	Statement
Form 369	<p> If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). </p>

Conclusion	Statement
Form 280	<p> There is a complete separable metric space with a subset which does not have the Baire property. </p>

The conclusion Form 369 \( \not \Rightarrow \) Form 313 then follows.

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

Name	Statement
\(\cal M5(\aleph)\) 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 \(\Bbb R\)