Implications

This non-implication, Form 150 \( \not \Rightarrow \) Form 427, 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: 9939, whose string of implications is:
    8 \(\Rightarrow\) 150
  • A proven non-implication whose code is 3. In this case, it's Code 3: 274, Form 8 \( \not \Rightarrow \) Form 144 whose summary information is:
    Hypothesis Statement
    Form 8 <p> \(C(\aleph_{0},\infty)\): </p>

    Conclusion Statement
    Form 144 <p> Every set is almost well orderable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5249, whose string of implications is:
    427 \(\Rightarrow\) 67 \(\Rightarrow\) 144

The conclusion Form 150 \( \not \Rightarrow \) Form 427 then follows.

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

Name Statement
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N38\) Howard/Rubin Model I Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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