Implications

This non-implication, Form 207-alpha \( \not \Rightarrow \) Form 267, 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: 10310, whose string of implications is:
    23 \(\Rightarrow\) 207-alpha
  • A proven non-implication whose code is 5. In this case, it's Code 3: 62, Form 23 \( \not \Rightarrow \) Form 267 whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </p>

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

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

The conclusion Form 207-alpha \( \not \Rightarrow \) Form 267 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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