Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 100, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10896, whose string of implications is:
    128 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 331, Form 128 \( \not \Rightarrow \) Form 376 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 376 <p> <strong>Restricted Kinna Wagner Principle</strong>:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 878, whose string of implications is:
    100 \(\Rightarrow\) 9 \(\Rightarrow\) 376

The conclusion Form 0 \( \not \Rightarrow \) Form 100 then follows.

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

Name Statement
\(\cal N3\) 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 \(A\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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