Axiom Of Choice

This non-implication, Form 134 \( \not \Rightarrow \) Form 53, 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: 361, Form 134 \( \not \Rightarrow \) Form 53 whose summary information is:

Hypothesis	Statement
Form 134	<p> If \(X\) is an infinite \(T_1\) space and \(X^{Y}\) is \(T_5\), then \(Y\) is countable. (\(T_5\) is 'hereditarily \(T_4\)'.) </p>

Conclusion	Statement
Form 53	<p> For all infinite cardinals \(m\), \(m^2\le 2^m\). <a href="/articles/Mathias-1979">Mathias [1979]</a>, prob 1336. </p>

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

The conclusion Form 134 \( \not \Rightarrow \) Form 53 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 \(A\)