Implications

This non-implication, Form 251 \( \not \Rightarrow \) Form 193, 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: 6424, whose string of implications is:
    89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 251
  • A proven non-implication whose code is 5. In this case, it's Code 3: 173, Form 89 \( \not \Rightarrow \) Form 64 whose summary information is:
    Hypothesis Statement
    Form 89 <p> <strong>Antichain Principle:</strong>  Every partially ordered set has a maximal antichain. <a href="/books/8">Jech [1973b]</a>, p 133. </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: 7677, whose string of implications is:
    193 \(\Rightarrow\) 188 \(\Rightarrow\) 106 \(\Rightarrow\) 126 \(\Rightarrow\) 82 \(\Rightarrow\) 83 \(\Rightarrow\) 64

The conclusion Form 251 \( \not \Rightarrow \) Form 193 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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