Implications

This non-implication, Form 209 \( \not \Rightarrow \) Form 101, 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: 10166, whose string of implications is:
    31 \(\Rightarrow\) 209
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1416, Form 31 \( \not \Rightarrow \) Form 369 whose summary information is:
    Hypothesis Statement
    Form 31 <p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong>  The union of a denumerable set of denumerable sets is denumerable. </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: 7083, whose string of implications is:
    101 \(\Rightarrow\) 100 \(\Rightarrow\) 369

The conclusion Form 209 \( \not \Rightarrow \) Form 101 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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