Implications

This non-implication, Form 128 \( \not \Rightarrow \) Form 179-epsilon, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1056, Form 128 \( \not \Rightarrow \) Form 144 whose summary information is:
    Hypothesis Statement
    Form 128 <p> <strong>Aczel's Realization Principle:</strong> On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. </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 128 \( \not \Rightarrow \) Form 179-epsilon then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)
\(\cal N38\) 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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