Axiom Of Choice

This non-implication, Form 376 $ \not \Rightarrow $ Form 99, 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: 10020, whose string of implications is: 15 $\Rightarrow$ 376

A proven non-implication whose code is 3. In this case, it's Code 3: 955, Form 15 $ \not \Rightarrow $ Form 99 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 99	<p> <strong>Rado's Selection Lemma:</strong> Let $\{K(\lambda): \lambda \in\Lambda\}$ be a family of finite subsets (of $X$) and suppose for each finite $S\subseteq\Lambda$ there is a function $\gamma(S): S \rightarrow X$ such that $(\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))$. Then there is an $f: \Lambda\rightarrow X$ such that for every finite $S\subseteq\Lambda$ there is a finite $T$ such that $S\subseteq T\subseteq\Lambda$ and such that $f$ and $\gamma (T)$ agree on S. </p>

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

The conclusion Form 376 $ \not \Rightarrow $ Form 99 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 $ZF$, $\cal M3$