Axiom Of Choice

This non-implication, Form 367 \( \not \Rightarrow \) Form 131, 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: 6812, whose string of implications is: 164 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 367

A proven non-implication whose code is 5. In this case, it's Code 3: 438, Form 164 \( \not \Rightarrow \) Form 131 whose summary information is:

Hypothesis	Statement
Form 164	<p> Every non-well-orderable set has an infinite subset with a Dedekind finite power set. </p>

Conclusion	Statement
Form 131	<p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

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

The conclusion Form 367 \( \not \Rightarrow \) Form 131 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