Implications

This non-implication, Form 119 \( \not \Rightarrow \) Form 239, whose code is 4, 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: 2308, whose string of implications is:
    31 \(\Rightarrow\) 32 \(\Rightarrow\) 119
  • A proven non-implication whose code is 3. In this case, it's Code 3: 125, Form 31 \( \not \Rightarrow \) Form 13 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 13 <p> Every Dedekind finite subset of \({\Bbb R}\) is finite. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7929, whose string of implications is:
    239 \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 77 \(\Rightarrow\) 185 \(\Rightarrow\) 13

The conclusion Form 119 \( \not \Rightarrow \) Form 239 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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