Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 175, 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: 165, Form 0 \( \not \Rightarrow \) Form 199(\(n\)) whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 199(\(n\)) <p> (For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1287, whose string of implications is:
    175 \(\Rightarrow\) 13 \(\Rightarrow\) 199(\(n\))

The conclusion Form 0 \( \not \Rightarrow \) Form 175 then follows.

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

Name Statement
\(\cal M26\) Kanovei's Model I Starting with a model of \(ZF + V = L\) and using forcing techniques due to <a href="/excerpts/Jensen-1968">Jensen [1968]</a>, Kanovei constructs a model of \(ZF\) in which there is an infinite Dedekind finite set \(A\) of generic reals that is in the class \(\varPi^1_n\), but there are no infinite Dedekind finite subsets of \(\Bbb R\) in the class \(\varSigma^1_n\), where \(n\in\omega\), \(n\ge 2\)

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