Axiom Of Choice

This non-implication, Form 191 \( \not \Rightarrow \) Form 293, 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: 510, Form 191 \( \not \Rightarrow \) Form 293 whose summary information is:

Hypothesis	Statement
Form 191	<p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

Conclusion	Statement
Form 293	<p> For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. </p>

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

The conclusion Form 191 \( \not \Rightarrow \) Form 293 then follows.

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

Name	Statement
\(\cal N37\) A variation of Blass' model, \(\cal N28\)	Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)