Implications

This non-implication, Form 242 \( \not \Rightarrow \) Form 408, 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: 10231, whose string of implications is:
    233 \(\Rightarrow\) 242
  • A proven non-implication whose code is 5. In this case, it's Code 3: 554, Form 233 \( \not \Rightarrow \) Form 64 whose summary information is:
    Hypothesis Statement
    Form 233 <p> <strong>Artin-Schreier theorem:</strong> If a field has an algebraic closure it is unique up to isomorphism. </p>

    Conclusion Statement
    Form 64 <p> \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4514, whose string of implications is:
    408 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n \(\Rightarrow\) 64

The conclusion Form 242 \( \not \Rightarrow \) Form 408 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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