Axiom Of Choice

This non-implication, Form 16 \( \not \Rightarrow \) Form 278, 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: 26, Form 16 \( \not \Rightarrow \) Form 278 whose summary information is:

Hypothesis	Statement
Form 16	<p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function. </p>

Conclusion	Statement
Form 278	<p> In an integral domain \(R\), if every ideal is finitely generated then \(R\) has a maximal proper ideal. <a href="/notes/note-45">note 45</a> E. </p>

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

The conclusion Form 16 \( \not \Rightarrow \) Form 278 then follows.

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

Name	Statement
\(\cal N1\) The Basic Fraenkel Model	The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)