Axiom Of Choice

This non-implication, Form 34 \( \not \Rightarrow \) Form 406, 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: 6933, whose string of implications is: 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 34

A proven non-implication whose code is 5. In this case, it's Code 3: 679, Form 337 \( \not \Rightarrow \) Form 10 whose summary information is:

Hypothesis	Statement
Form 337	<p> \(C(WO\), <strong>uniformly linearly ordered</strong>): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </p>

Conclusion	Statement
Form 10	<p> \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9595, whose string of implications is: 406 \(\Rightarrow\) 10

The conclusion Form 34 \( \not \Rightarrow \) Form 406 then follows.

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

Name	Statement
\(\cal N2\) The Second Fraenkel Model	The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).
\(\cal N2(n)\) A generalization of \(\cal N2\)	This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\)
\(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\)	\(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)
\(\cal N6\) Levy's Model I	\(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(\|P_n\| = p_n\), where \(p_n\) is the \(n\)th prime
\(\cal N50(E)\) Brunner's Model III	\(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)

Implications