Implications

This non-implication, Form 119 \( \not \Rightarrow \) Form 253, 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: 10175, whose string of implications is:
    118 \(\Rightarrow\) 119
  • A proven non-implication whose code is 3. In this case, it's Code 3: 91, Form 118 \( \not \Rightarrow \) Form 253 whose summary information is:
    Hypothesis Statement
    Form 118 <p> Every linearly orderable topological space is normal.  <a href="/books/28">Birkhoff [1967]</a>, p 241. </p>

    Conclusion Statement
    Form 253 <p> <strong>\L o\'s' Theorem:</strong> If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 119 \( \not \Rightarrow \) Form 253 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