Axiom Of Choice

This non-implication, Form 379 \( \not \Rightarrow \) Form 128, 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: 725, Form 379 \( \not \Rightarrow \) Form 128 whose summary information is:

Hypothesis	Statement
Form 379	<p> \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). </p>

Conclusion	Statement
Form 128	<p> <strong>Aczel's Realization Principle:</strong> On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. </p>

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

The conclusion Form 379 \( \not \Rightarrow \) Form 128 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 \(A\)