Implications

This non-implication, Form 144 \( \not \Rightarrow \) Form 90, 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: 10307, whose string of implications is:
    179-epsilon \(\Rightarrow\) 144
  • A proven non-implication whose code is 3. In this case, it's Code 3: 939, Form 179-epsilon \( \not \Rightarrow \) Form 91 whose summary information is:
    Hypothesis Statement
    Form 179-epsilon <p> Suppose  \(\epsilon > 0\) is an ordinal. \(\forall x\), \(x\in W(\epsilon\)). </p>

    Conclusion Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9779, whose string of implications is:
    90 \(\Rightarrow\) 91

The conclusion Form 144 \( \not \Rightarrow \) Form 90 then follows.

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

Name Statement
\(\cal M35(\epsilon)\) David's Model In Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a>, define sets \(B_n=\{x\subset\omega: |x\ \Delta\ a_n| <\omega\vee |x\ \Delta\ \omega-a_n| \le\omega\}\) (where \(\Delta\) is the symmetric difference)

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