Axiom Of Choice

This non-implication, Form 124 $\not \Rightarrow$ Form 424, 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: 77, whose string of implications is: 3 $\Rightarrow$ 9 $\Rightarrow$ 17 $\Rightarrow$ 124

A proven non-implication whose code is 3. In this case, it's Code 3: 235, Form 3

$\not \Rightarrow$ Form 5 whose summary information is:

Hypothesis	Statement
Form 3	$2m = m$ : For all infinite cardinals $m$ , $2m = m$ .

Conclusion	Statement
Form 5	<p> $C(\aleph_0,\aleph_0,\Bbb R)$ : Every denumerable set of non-empty denumerable subsets of ${\Bbb R}$ has a choice function. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6963, whose string of implications is: 424 $\Rightarrow$ 94 $\Rightarrow$ 5

The conclusion Form 124 $\not \Rightarrow$ Form 424 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 $ZF + V = L$