Axiom Of Choice

This non-implication, Form 390 $ \not \Rightarrow $ Form 218, 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: 1683, whose string of implications is: 15 $\Rightarrow$ 30 $\Rightarrow$ 62 $\Rightarrow$ 64 $\Rightarrow$ 390

A proven non-implication whose code is 3. In this case, it's Code 3: 1225, Form 15 $ \not \Rightarrow $ Form 289 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 289	<p> If $S$ is a set of subsets of a countable set and $S$ is closed under chain unions, then $S$ has a $\subseteq$-maximal element. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6093, whose string of implications is: 218 $\Rightarrow$ 67 $\Rightarrow$ 89 $\Rightarrow$ 90 $\Rightarrow$ 91 $\Rightarrow$ 79 $\Rightarrow$ 289

The conclusion Form 390 $ \not \Rightarrow $ Form 218 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