Implications

This non-implication, Form 31 \( \not \Rightarrow \) Form 90, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1226, Form 31 \( \not \Rightarrow \) Form 289 whose summary information is:
    Hypothesis Statement
    Form 31 <p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong>  The union of a denumerable set of denumerable sets is denumerable. </p>

    Conclusion Statement
    Form 289 <p> If \(S\) is a set of subsets of a countable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6569, whose string of implications is:
    90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 289

The conclusion Form 31 \( \not \Rightarrow \) Form 90 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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