Axiom Of Choice

This non-implication, Form 108 \( \not \Rightarrow \) Form 302, 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: 6940, whose string of implications is: 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 5 \(\Rightarrow\) 38 \(\Rightarrow\) 108

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: 8909, whose string of implications is: 302 \(\Rightarrow\) 392 \(\Rightarrow\) 393 \(\Rightarrow\) 121 \(\Rightarrow\) 122 \(\Rightarrow\) 10

The conclusion Form 108 \( \not \Rightarrow \) Form 302 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