Implications

This non-implication, Form 411 \( \not \Rightarrow \) Form 175, 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: 1642, whose string of implications is:
    14 \(\Rightarrow\) 410 \(\Rightarrow\) 411
  • 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: 10177, whose string of implications is:
    175 \(\Rightarrow\) 13

The conclusion Form 411 \( \not \Rightarrow \) Form 175 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