Axiom Of Choice

This non-implication, Form 134 \( \not \Rightarrow \) Form 168, 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: 367, Form 134 \( \not \Rightarrow \) Form 126 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 126	<p> \(MC(\aleph_0,\infty)\), <strong>Countable axiom of multiple choice:</strong> For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7193, whose string of implications is: 168 \(\Rightarrow\) 100 \(\Rightarrow\) 347 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 126

The conclusion Form 134 \( \not \Rightarrow \) Form 168 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 \(A\)