Implications

This non-implication, Form 216 \( \not \Rightarrow \) Form 1, 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: 10311, whose string of implications is:
    217 \(\Rightarrow\) 216
  • A proven non-implication whose code is 5. In this case, it's Code 3: 539, Form 217 \( \not \Rightarrow \) Form 267 whose summary information is:
    Hypothesis Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

    Conclusion Statement
    Form 267 <p> There is no infinite, free complete Boolean algebra. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10580, whose string of implications is:
    1 \(\Rightarrow\) 267

The conclusion Form 216 \( \not \Rightarrow \) Form 1 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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