Axiom Of Choice

This non-implication, Form 190 $\not \Rightarrow$ Form 286, 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: 7695, whose string of implications is: 191 $\Rightarrow$ 189 $\Rightarrow$ 190

A proven non-implication whose code is 3. In this case, it's Code 3: 16, Form 191

$\not \Rightarrow$ Form 65 whose summary information is:

Hypothesis	Statement
Form 191	<p> $SVC$ : There is a set $S$ such that for every set $a$ , there is an ordinal $\alpha$ and a function from $S\times\alpha$ onto $a$ . </p>

Conclusion	Statement
Form 65	<p> <strong>The Krein-Milman Theorem:</strong> Let $K$ be a compact convex set in a locally convex topological vector space $X$ . Then $K$ has an extreme point. (An <em>extreme point</em> is a point which is not an interior point of any line segment which lies in $K$ .) <a href="/books/23">Rubin, H./Rubin, J. [1985]</a> p. 177. <p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9608, whose string of implications is: 286 $\Rightarrow$ 65

The conclusion Form 190 $\not \Rightarrow$ Form 286 then follows.

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

Name	Statement
$\cal M1$ Cohen's original model	Add a denumerable number of generic reals (subsets of $\omega$ ), $a_1$ , $a_2$ , $\cdots$ , along with the set $b$ containing them