Axiom Of Choice

This non-implication, Form 369 \( \not \Rightarrow \) Form 23, 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: 117, Form 369 \( \not \Rightarrow \) Form 34 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 34 <p> \(\aleph_{1}\) is regular. </p>
An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1880, whose string of implications is: 23 \(\Rightarrow\) 25 \(\Rightarrow\) 34

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 34	<p> \(\aleph_{1}\) is regular. </p>

The conclusion Form 369 \( \not \Rightarrow \) Form 23 then follows.

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

Name	Statement
\(\cal M12(\aleph)\) 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