Axiom Of Choice

This non-implication, Form 355 $ \not \Rightarrow $ Form 86-alpha, 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: 1698, whose string of implications is: 15 $\Rightarrow$ 322 $\Rightarrow$ 355

A proven non-implication whose code is 3. In this case, it's Code 3: 156, Form 15 $ \not \Rightarrow $ Form 9 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 9	<p>Finite $\Leftrightarrow$ Dedekind finite: $W_{\aleph_{0}}$ <a href="/books/8">Jech [1973b]</a>: $E(I,IV)$ <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 367, whose string of implications is: 86-alpha $\Rightarrow$ 8 $\Rightarrow$ 9

The conclusion Form 355 $ \not \Rightarrow $ Form 86-alpha 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