Implications

This non-implication, Form 191 \( \not \Rightarrow \) Form 392, 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: 18, Form 191 \( \not \Rightarrow \) Form 13 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 13 <p> Every Dedekind finite subset of \({\Bbb R}\) is finite. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9522, whose string of implications is:
    392 \(\Rightarrow\) 394 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 13

The conclusion Form 191 \( \not \Rightarrow \) Form 392 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

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