Axiom Of Choice

This non-implication, Form 283 $ \not \Rightarrow $ Form 303, 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: 1676, whose string of implications is: 15 $\Rightarrow$ 30 $\Rightarrow$ 62 $\Rightarrow$ 283

A proven non-implication whose code is 3. In this case, it's Code 3: 126, Form 15 $ \not \Rightarrow $ Form 14 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 14	<p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4051, whose string of implications is: 303 $\Rightarrow$ 50 $\Rightarrow$ 14

The conclusion Form 283 $ \not \Rightarrow $ Form 303 then follows.

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

Name	Statement
$\cal M3$ Mathias' model	Mathias proves that the $FM$ model <a href="/models/Mathias-Pincus-1">$\cal N4$</a> can be transformed into a model of $ZF$, $\cal M3$