Axiom Of Choice

This non-implication, Form 142 $\not \Rightarrow$ Form 305, whose code is 4, is constructed around a proven non-implication as follows:

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

A proven non-implication whose code is 3. In this case, it's Code 3: 239, Form 142

$\not \Rightarrow$ Form 93 whose summary information is:

Hypothesis	Statement
Form 142	<p> $\neg PB$ : There is a set of reals without the property of Baire. <a href="/books/8">Jech [1973b]</a>, p. 7. </p>

Conclusion	Statement
Form 93	<p> There is a non-measurable subset of ${\Bbb R}$ . </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8931, whose string of implications is: 305 $\Rightarrow$ 306 $\Rightarrow$ 93

The conclusion Form 142 $\not \Rightarrow$ Form 305 then follows.

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

Name	Statement
$\cal M38$ Shelah's Model II	In a model of $ZFC +$ " $\kappa$ is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse $\kappa$ to $\aleph_1$ similarly to <a href="/articles/Solovay-1970">Solovay [1970]</a>