This non-implication,
Form 6 ⇏
Form 1,
whose code is 4, is constructed around a proven non-implication as follows:
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> |
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 |