Axiom Of Choice

This non-implication, Form 362 \( \not \Rightarrow \) Form 84, 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: 6900, whose string of implications is: 164 \(\Rightarrow\) 91 \(\Rightarrow\) 361 \(\Rightarrow\) 362

A proven non-implication whose code is 5. In this case, it's Code 3: 433, Form 164 \( \not \Rightarrow \) Form 84 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 84	<p> \(E(II,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): \((\forall x)(x\) is \(T\)-finite if and only if \(\cal P(x)\) is Dedekind finite). </p>

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

The conclusion Form 362 \( \not \Rightarrow \) Form 84 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