Axiom Of Choice

This non-implication, Form 356 $ \not \Rightarrow $ Form 392, 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: 1699, whose string of implications is: 15 $\Rightarrow$ 323 $\Rightarrow$ 356

A proven non-implication whose code is 3. In this case, it's Code 3: 1335, Form 15 $ \not \Rightarrow $ Form 330 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 330	<p> $MC(WO,WO)$: For every well ordered set $X$ of well orderable sets such that for all $x\in X$, $\|x\|\ge 1$, there is a function $f$ such that for every $x\in X$, $f(x)$ is a finite, non-empty subset of $x$. (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9517, whose string of implications is: 392 $\Rightarrow$ 393 $\Rightarrow$ 165 $\Rightarrow$ 330

The conclusion Form 356 $ \not \Rightarrow $ Form 392 then follows.

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

Name	Statement
$\cal M34(\aleph_1)$ Pincus' Model III	Pincus proves that Cohen's model <a href="/models/Cohen-1">$\cal M1$</a> can be extended by adding $\aleph_1$ generic sets along with the set $b$ containing them and well orderings of all countable subsets of $b$