Axiom Of Choice

This non-implication, Form 39 \( \not \Rightarrow \) Form 1, 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: 10106, whose string of implications is: 40 \(\Rightarrow\) 39

A proven non-implication whose code is 5. In this case, it's Code 3: 111, Form 40 \( \not \Rightarrow \) Form 192 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 192	<p> \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10505, whose string of implications is: 1 \(\Rightarrow\) 192

The conclusion Form 39 \( \not \Rightarrow \) Form 1 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 \(A\)
\(\cal N56\) Howard's model III: Assume the the atoms are indexed asfollows: \(A = \{a(i,j) : i\in{\Bbb Q} \hbox{ and } j\in\omega \}\)	Foreach \(i\in \Bbb Q\), let \(A_i = \{a(i,j) : j\in \omega\}\)

Implications