Implications

This non-implication, Form 315 \( \not \Rightarrow \) Form 258, whose code is 6, 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: 6479, whose string of implications is:
    112 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 25 \(\Rightarrow\) 315
  • A proven non-implication whose code is 5. In this case, it's Code 3: 259, Form 112 \( \not \Rightarrow \) Form 106 whose summary information is:
    Hypothesis Statement
    Form 112 <p> \(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). </p>

    Conclusion Statement
    Form 106 <p> <strong>Baire Category Theorem for Compact Hausdorff Spaces:</strong> Every compact Hausdorff space is Baire. <p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8360, whose string of implications is:
    258 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 43 \(\Rightarrow\) 106

The conclusion Form 315 \( \not \Rightarrow \) Form 258 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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