Axiom Of Choice

This non-implication, Form 114 \( \not \Rightarrow \) Form 76, 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: 276, Form 114 \( \not \Rightarrow \) Form 76 whose summary information is:

Hypothesis	Statement
Form 114	<p> Every A-bounded \(T_2\) topological space is weakly Loeb. (<em>\(A\)-bounded</em> means amorphous subsets are relatively compact. <em>Weakly Loeb</em> means the set of non-empty closed subsets has a multiple choice function.) </p>

Conclusion	Statement
Form 76	<p> \(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

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

The conclusion Form 114 \( \not \Rightarrow \) Form 76 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 \(A\)