Axiom Of Choice

This non-implication, Form 13 \( \not \Rightarrow \) Form 157, 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: 10130, whose string of implications is: 16 \(\Rightarrow\) 13

A proven non-implication whose code is 5. In this case, it's Code 3: 22, Form 16 \( \not \Rightarrow \) Form 157 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 157	<p> <strong>Theorem of Goodner:</strong> A compact \(T_{2}\) space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of \(C(X)\) (set of continuous real valued functions on \(X\)) which is pointwise bounded has a supremum. </p>

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

The conclusion Form 13 \( \not \Rightarrow \) Form 157 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