Axiom Of Choice

This non-implication, Form 198 $ \not \Rightarrow $ Form 144, 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: 1659, whose string of implications is: 15 $\Rightarrow$ 30 $\Rightarrow$ 198

A proven non-implication whose code is 3. In this case, it's Code 3: 1051, Form 15 $ \not \Rightarrow $ Form 144 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 144	<p> Every set is almost well orderable. </p>

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

The conclusion Form 198 $ \not \Rightarrow $ Form 144 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
$\cal M3$ Mathias' model	Mathias proves that the $FM$ model <a href="/models/Mathias-Pincus-1">$\cal N4$</a> can be transformed into a model of $ZF$, $\cal M3$
$\cal M11$ Forti/Honsell Model	Using a model of $ZF + V = L$ for the ground model, the authors construct a generic extension, $\cal M$, using Easton forcing which adds $\kappa$ generic subsets to each regular cardinal $\kappa$

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
\(\cal M3\) Mathias' model	Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)
\(\cal M11\) Forti/Honsell Model	Using a model of \(ZF + V = L\) for the ground model, the authors construct a generic extension, \(\cal M\), using Easton forcing which adds \(\kappa\) generic subsets to each regular cardinal \(\kappa\)