Implications

This non-implication, Form 197 \( \not \Rightarrow \) Form 156, 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: 5974, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 197
  • A proven non-implication whose code is 5. In this case, it's Code 3: 208, Form 91 \( \not \Rightarrow \) Form 156 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

    Conclusion Statement
    Form 156 <p> <strong>Theorem of Gelfand and Kolmogoroff:</strong> Two compact \(T_2\) spaces are  homeomorphic if their rings of real valued continuous functions are isomorphic. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 197 \( \not \Rightarrow \) Form 156 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 \(A\)

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