Implications

This non-implication, Form 111 \( \not \Rightarrow \) Form 363, 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: 7613, whose string of implications is:
    231 \(\Rightarrow\) 151 \(\Rightarrow\) 122 \(\Rightarrow\) 250 \(\Rightarrow\) 111
  • A proven non-implication whose code is 3. In this case, it's Code 3: 249, Form 231 \( \not \Rightarrow \) Form 273 whose summary information is:
    Hypothesis Statement
    Form 231 <p> \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable. </p>

    Conclusion Statement
    Form 273 <p> There is a subset of \({\Bbb R}\) which is not Borel. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9365, whose string of implications is:
    363 \(\Rightarrow\) 364 \(\Rightarrow\) 273

The conclusion Form 111 \( \not \Rightarrow \) Form 363 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M12(\aleph)\) Truss' Model I This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular

Edit | Back