Axiom Of Choice

This non-implication, Form 314 \( \not \Rightarrow \) Form 239, whose code is 4, is constructed around a proven non-implication as follows:

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4211, whose string of implications is: 60 \(\Rightarrow\) 85 \(\Rightarrow\) 314

A proven non-implication whose code is 3. In this case, it's Code 3: 932, Form 60 \( \not \Rightarrow \) Form 84 whose summary information is:

Hypothesis	Statement
Form 60	<p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </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: 7928, whose string of implications is: 239 \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 77 \(\Rightarrow\) 185 \(\Rightarrow\) 84

The conclusion Form 314 \( \not \Rightarrow \) Form 239 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 \(A\)
\(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\))	\(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\)