Implications

This non-implication, Form 277 \( \not \Rightarrow \) Form 394, 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: 863, Form 277 \( \not \Rightarrow \) Form 13 whose summary information is:
    Hypothesis Statement
    Form 277 <p> \(E(D,VII)\): Every non-well-orderable cardinal is decomposable. </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: 9303, whose string of implications is:
    394 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 13

The conclusion Form 277 \( \not \Rightarrow \) Form 394 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