Axiom Of Choice

This non-implication, Form 75 \( \not \Rightarrow \) Form 1, 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: 90, whose string of implications is: 3 \(\Rightarrow\) 4 \(\Rightarrow\) 405 \(\Rightarrow\) 75
A proven non-implication whose code is 3. In this case, it's Code 3: 237, Form 3 \( \not \Rightarrow \) Form 1 whose summary information is:

Hypothesis Statement

Form 3 \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

Conclusion Statement

Form 1 <p> \(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets has a choice function. </p>
This non-implication was constructed without the use of this last code 2/1 implication

Hypothesis	Statement
Form 3	\(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

Conclusion	Statement
Form 1	<p> \(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets has a choice function. </p>

The conclusion Form 75 \( \not \Rightarrow \) Form 1 then follows.

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

Name	Statement
\(\cal M6\) Sageev's Model I	Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)
\(\cal N9\) Halpern/Howard Model	\(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\)