Axiom Of Choice

This non-implication, Form 73 \( \not \Rightarrow \) Form 427, 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: 1726, whose string of implications is: 325 \(\Rightarrow\) 17 \(\Rightarrow\) 132 \(\Rightarrow\) 73

A proven non-implication whose code is 5. In this case, it's Code 3: 654, Form 325 \( \not \Rightarrow \) Form 84 whose summary information is:

Hypothesis	Statement
Form 325	<p> <strong>Ramsey's Theorem II:</strong> \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see <a href="/form-classes/howard-rubin-17">Form 17</a>.) </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>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5498, whose string of implications is: 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 77 \(\Rightarrow\) 185 \(\Rightarrow\) 84

The conclusion Form 73 \( \not \Rightarrow \) Form 427 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