Axiom Of Choice

This non-implication, Form 40 \( \not \Rightarrow \) Form 193, whose code is 6, is constructed around a proven non-implication as follows:

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

A proven non-implication whose code is 5. In this case, it's Code 3: 112, Form 40 \( \not \Rightarrow \) Form 193 whose summary information is:

Hypothesis	Statement
Form 40	<p> \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </p>

Conclusion	Statement
Form 193	<p> \(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group. </p>

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

The conclusion Form 40 \( \not \Rightarrow \) Form 193 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name	Statement
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\)	Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\)	Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above
\(\cal N33\) Howard/H\.Rubin/J\.Rubin 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 boundedsubsets of \(A\)