Axiom Of Choice

This non-implication, Form 199(\(n\)) \( \not \Rightarrow \) Form 177, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1286, whose string of implications is: 16 \(\Rightarrow\) 13 \(\Rightarrow\) 199(\(n\))

A proven non-implication whose code is 5. In this case, it's Code 3: 23, Form 16 \( \not \Rightarrow \) Form 177 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 177	<p> An infinite box product of regular \(T_1\) spaces, each of cardinality greater than 1, is neither first countable nor connected. </p>

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

The conclusion Form 199(\(n\)) \( \not \Rightarrow \) Form 177 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 \(A\)

Implications