Axiom Of Choice

This non-implication, Form 243 \( \not \Rightarrow \) Form 239, 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: 3905, whose string of implications is: 44 \(\Rightarrow\) 43 \(\Rightarrow\) 243

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: 7880, whose string of implications is: 239 \(\Rightarrow\) 427 \(\Rightarrow\) 67

The conclusion Form 243 \( \not \Rightarrow \) Form 239 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