Axiom Of Choice

This non-implication, Form 198 $ \not \Rightarrow $ Form 15, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 65, whose string of implications is: 3 $\Rightarrow$ 9 $\Rightarrow$ 198

A proven non-implication whose code is 5. In this case, it's Code 3: 2, Form 3 $ \not \Rightarrow $ Form 15 whose summary information is:

Hypothesis	Statement
Form 3	$2m = m$: For all infinite cardinals $m$, $2m = m$.

Conclusion	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>

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

The conclusion Form 198 $ \not \Rightarrow $ Form 15 then follows.

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

Name	Statement
$\cal N9$ Halpern/Howard Model	$A$ is a set of atoms with the structureof the set $ \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}$

Implications