Implications

This non-implication, Form 355 \( \not \Rightarrow \) Form 179-epsilon, 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: 10004, whose string of implications is:
    8 \(\Rightarrow\) 355
  • 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: 10307, whose string of implications is:
    179-epsilon \(\Rightarrow\) 144

The conclusion Form 355 \( \not \Rightarrow \) Form 179-epsilon 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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