Implications

This non-implication, Form 137-k \( \not \Rightarrow \) Form 90, 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: 1505, whose string of implications is:
    14 \(\Rightarrow\) 139 \(\Rightarrow\) 137-k
  • A proven non-implication whose code is 3. In this case, it's Code 3: 155, Form 14 \( \not \Rightarrow \) Form 79 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

    Conclusion Statement
    Form 79 <p> \({\Bbb R}\) can be well ordered.  <a href="/articles/hilbert-1900">Hilbert [1900]</a>, p 263. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6552, whose string of implications is:
    90 \(\Rightarrow\) 91 \(\Rightarrow\) 79

The conclusion Form 137-k \( \not \Rightarrow \) 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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