Axiom Of Choice

This non-implication, Form 6 $\not \Rightarrow$ Form 1, 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: 786, Form 6

$\not \Rightarrow$ Form 158 whose summary information is:

Hypothesis	Statement
Form 6	<p> $UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)$ : The union of a denumerable family of denumerable subsets of ${\Bbb R}$ is denumerable. </p>

Conclusion	Statement
Form 158	<p> In every Hilbert space $H$ , if the closed unit ball is sequentially compact, then $H$ has an orthonormal basis. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10471, whose string of implications is: 1 $\Rightarrow$ 158

The conclusion Form 6 $\not \Rightarrow$ Form 1 then follows.

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

Name	Statement
$\cal N25$ Brunner's Model I	The set of atoms, $A$ , is equipped with thestructure of the Hilbert space $\ell_2$ , $\cal G$ is the group of allpermutations on $A$ that preserve the norm (unitary operators), and $S$ isthe set of all finite subsets of $A$