Axiom Of Choice

This non-implication, Form 31 \( \not \Rightarrow \) Form 355, 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: 1708, whose string of implications is: 16 \(\Rightarrow\) 352 \(\Rightarrow\) 31

A proven non-implication whose code is 5. In this case, it's Code 3: 30, Form 16 \( \not \Rightarrow \) Form 355 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 355	<p> \(KW(\aleph_0,\infty)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of sets: For every denumerable set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

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

The conclusion Form 31 \( \not \Rightarrow \) Form 355 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 \(A\)
\(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\)	The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\)

Implications