Implications

This non-implication, Form 327 \( \not \Rightarrow \) Form 325, 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: 7464, whose string of implications is:
    121 \(\Rightarrow\) 122 \(\Rightarrow\) 327
  • A proven non-implication whose code is 3. In this case, it's Code 3: 987, Form 121 \( \not \Rightarrow \) Form 124 whose summary information is:
    Hypothesis Statement
    Form 121 <p> \(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function. </p>

    Conclusion Statement
    Form 124 <p> Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and  a  scalar operator.  (A set is <em>amorphous</em> if it is not the union of two disjoint infinite sets.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1710, whose string of implications is:
    325 \(\Rightarrow\) 17 \(\Rightarrow\) 124

The conclusion Form 327 \( \not \Rightarrow \) Form 325 then follows.

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

Name Statement
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)

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