Axiom Of Choice

This non-implication, Form 345 \( \not \Rightarrow \) Form 193, whose code is 6, 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 5. In this case, it's Code 3: 709, Form 345 \( \not \Rightarrow \) Form 193 whose summary information is:

Hypothesis	Statement
Form 345	<p> <strong>Rasiowa-Sikorski Axiom:</strong> If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). </p>

Conclusion	Statement
Form 193	<p> \(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group. </p>

This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 345 \( \not \Rightarrow \) Form 193 then follows.

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

Name	Statement
\(\cal N40\) Howard/Rubin Model II	A variation of \(\cal N38\)