Implications

This non-implication, Form 125 \( \not \Rightarrow \) Form 334, 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: 7604, whose string of implications is:
    179-epsilon \(\Rightarrow\) 144 \(\Rightarrow\) 125
  • 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: 5413, whose string of implications is:
    334 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91

The conclusion Form 125 \( \not \Rightarrow \) Form 334 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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