Implications

This non-implication, Form 82 \( \not \Rightarrow \) Form 126, 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: 150, Form 82 \( \not \Rightarrow \) Form 13 whose summary information is:
    Hypothesis Statement
    Form 82 <p> \(E(I,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) </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: 6985, whose string of implications is:
    126 \(\Rightarrow\) 94 \(\Rightarrow\) 13

The conclusion Form 82 \( \not \Rightarrow \) Form 126 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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