Axiom Of Choice

This non-implication, Form 77 \( \not \Rightarrow \) Form 66, whose code is 4, 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: 3906, whose string of implications is: 44 \(\Rightarrow\) 43 \(\Rightarrow\) 77

A proven non-implication whose code is 3. In this case, it's Code 3: 924, Form 44 \( \not \Rightarrow \) Form 67 whose summary information is:

Hypothesis	Statement
Form 44	<p> \(DC(\aleph _{1})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(\|Y\| < \aleph_{1}\) there is an \(x \in X\) with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\). </p>

Conclusion	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>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9619, whose string of implications is: 66 \(\Rightarrow\) 67

The conclusion Form 77 \( \not \Rightarrow \) Form 66 then follows.

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

Name	Statement
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\)	Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above