Axiom Of Choice

This non-implication, Form 329 \( \not \Rightarrow \) Form 359, whose code is 6, 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: 9975, whose string of implications is: 67 \(\Rightarrow\) 329

A proven non-implication whose code is 5. In this case, it's Code 3: 155, Form 67 \( \not \Rightarrow \) Form 80 whose summary information is:

Hypothesis	Statement
Form 67	<p> \(MC(\infty,\infty)\) \((MC)\), <strong>The Axiom of Multiple Choice:</strong> For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). </p>

Conclusion	Statement
Form 80	<p> \(C(\aleph_{0},2)\): Every denumerable set of pairs has a choice function. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1798, whose string of implications is: 359 \(\Rightarrow\) 20 \(\Rightarrow\) 21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80

The conclusion Form 329 \( \not \Rightarrow \) Form 359 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 \( A \).