Implications

This non-implication, Form 182 \( \not \Rightarrow \) Form 41, 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: 9618, whose string of implications is:
    191 \(\Rightarrow\) 182
  • 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: 900, whose string of implications is:
    41 \(\Rightarrow\) 9 \(\Rightarrow\) 13

The conclusion Form 182 \( \not \Rightarrow \) Form 41 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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