Axiom Of Choice

This non-implication, Form 401 $ \not \Rightarrow $ Form 255, 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: 1685, whose string of implications is: 15 $\Rightarrow$ 30 $\Rightarrow$ 62 $\Rightarrow$ 121 $\Rightarrow$ 401

A proven non-implication whose code is 3. In this case, it's Code 3: 1031, Form 15 $ \not \Rightarrow $ Form 131 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 131	<p> $MC_\omega(\aleph_0,\infty)$: For every denumerable family $X$ of pairwise disjoint non-empty sets, there is a function $f$ such that for each $x\in X$, f(x) is a non-empty countable subset of $x$. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8085, whose string of implications is: 255 $\Rightarrow$ 260 $\Rightarrow$ 40 $\Rightarrow$ 39 $\Rightarrow$ 8 $\Rightarrow$ 126 $\Rightarrow$ 131

The conclusion Form 401 $ \not \Rightarrow $ Form 255 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