Implications

This non-implication, Form 190 \( \not \Rightarrow \) Form 100, 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: 7695, whose string of implications is:
    191 \(\Rightarrow\) 189 \(\Rightarrow\) 190
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1422, Form 191 \( \not \Rightarrow \) Form 369 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 369 <p> If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9832, whose string of implications is:
    100 \(\Rightarrow\) 369

The conclusion Form 190 \( \not \Rightarrow \) Form 100 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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