Implications

This non-implication, Form 380 \( \not \Rightarrow \) Form 157, 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: 10025, whose string of implications is:
    60 \(\Rightarrow\) 380
  • A proven non-implication whose code is 3. In this case, it's Code 3: 29, Form 60 \( \not \Rightarrow \) Form 157 whose summary information is:
    Hypothesis Statement
    Form 60 <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </p>

    Conclusion Statement
    Form 157 <p> <strong>Theorem of Goodner:</strong> A compact \(T_{2}\) space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of \(C(X)\) (set of continuous real valued functions on \(X\)) which is pointwise bounded has a supremum. </p>

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

The conclusion Form 380 \( \not \Rightarrow \) Form 157 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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