Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 337, 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: 164, Form 0 \( \not \Rightarrow \) Form 194 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 194 <p> \(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). <a href="/excerpts/Kanovei-1979">Kanovei [1979]</a>. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6937, whose string of implications is:
    337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 194

The conclusion Form 0 \( \not \Rightarrow \) Form 337 then follows.

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

Name Statement
\(\cal M41\) Kanovei's Model III Let \(\Bbb P\) be the set of conditions from the model in <a href="/excerpts/Jensen-1968">Jensen [1968]</a>

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