Implications

This non-implication, Form 38 \( \not \Rightarrow \) Form 158, 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: 9751, whose string of implications is:
    35 \(\Rightarrow\) 38
  • A proven non-implication whose code is 3. In this case, it's Code 3: 439, Form 35 \( \not \Rightarrow \) Form 158 whose summary information is:
    Hypothesis Statement
    Form 35 <p> The union of countably many meager subsets of \({\Bbb R}\) is meager. (Meager sets are the same as sets of the first category.) <a href="/books/8">Jech [1973b]</a> p 7 prob 1.7. </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>

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

The conclusion Form 38 \( \not \Rightarrow \) Form 158 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 \(A\)

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