Implications

This non-implication, Form 26 \( \not \Rightarrow \) Form 124, whose code is 6, 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: 9620, whose string of implications is:
    24 \(\Rightarrow\) 26
  • A proven non-implication whose code is 5. In this case, it's Code 3: 83, Form 24 \( \not \Rightarrow \) Form 124 whose summary information is:
    Hypothesis Statement
    Form 24 <p> \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) 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>

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

The conclusion Form 26 \( \not \Rightarrow \) Form 124 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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