Axiom Of Choice

This non-implication, Form 296 $ \not \Rightarrow $ Form 369, 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: 9862, whose string of implications is: 15 $\Rightarrow$ 296

A proven non-implication whose code is 3. In this case, it's Code 3: 1415, Form 15 $ \not \Rightarrow $ Form 369 whose summary information is:

Hypothesis	Statement
Form 15	<p> $KW(\infty,\infty)$ (KW), <strong>The Kinna-Wagner Selection Principle:</strong> For every set $M$ there is a function $f$ such that for all $A\in M$, if $\|A\|>1$ then $\emptyset\neq f(A)\subsetneq A$. (See <a href="/form-classes/howard-rubin-81($n$)">Form 81($n$)</a>. </p>

Conclusion	Statement
Form 369	<p> If $\Bbb R$ is partitioned into two sets, at least one of them has cardinality $2^{\aleph_0}$. </p>

This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 296 $ \not \Rightarrow $ Form 369 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