Axiom Of Choice

This non-implication, Form 396 \( \not \Rightarrow \) Form 62, 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: 5842, whose string of implications is: 67 \(\Rightarrow\) 112 \(\Rightarrow\) 395 \(\Rightarrow\) 396

A proven non-implication whose code is 5. In this case, it's Code 3: 163, Form 67 \( \not \Rightarrow \) Form 342-n 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 342-n	<p> (For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\): Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See <a href="/form-classes/howard-rubin-166">Form 166</a>.) </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4241, whose string of implications is: 62 \(\Rightarrow\) 61 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 73 \(\Rightarrow\) 342-n

The conclusion Form 396 \( \not \Rightarrow \) Form 62 then follows.

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

Name	Statement
\(\cal N50(E)\) 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 \(p^n\)