Implications

This non-implication, Form 140 \( \not \Rightarrow \) Form 133, 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: 4010, whose string of implications is:
    14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 140
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1224, Form 14 \( \not \Rightarrow \) Form 289 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </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: 6603, whose string of implications is:
    133 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 289

The conclusion Form 140 \( \not \Rightarrow \) Form 133 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

Edit | Back