Implications

This non-implication, Form 121 \( \not \Rightarrow \) Form 261, 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: 11581, whose string of implications is:
    121 \(\Rightarrow\) 121
  • 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: 8570, whose string of implications is:
    261 \(\Rightarrow\) 256 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 17 \(\Rightarrow\) 124

The conclusion Form 121 \( \not \Rightarrow \) Form 261 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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