Axiom Of Choice

This non-implication, Form 74 \( \not \Rightarrow \) Form 304, 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: 1707, whose string of implications is: 16 \(\Rightarrow\) 94 \(\Rightarrow\) 74

A proven non-implication whose code is 5. In this case, it's Code 3: 29, Form 16 \( \not \Rightarrow \) Form 304 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 304	<p> There does not exist a \(T_2\) topological space \(X\) such that every infinite subset of \(X\) contains an infinite compact subset. </p>

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

The conclusion Form 74 \( \not \Rightarrow \) Form 304 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name	Statement
\(\cal N3\) Mostowski's Linearly Ordered Model	\(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

Implications