Implications

This non-implication, Form 45-n \( \not \Rightarrow \) Form 89, 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: 3991, whose string of implications is:
    14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 45-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 152, Form 14 \( \not \Rightarrow \) Form 13 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </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: 6402, whose string of implications is:
    89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 77 \(\Rightarrow\) 185 \(\Rightarrow\) 13

The conclusion Form 45-n \( \not \Rightarrow \) Form 89 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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