Implications

This non-implication, Form 209 \( \not \Rightarrow \) Form 203, 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: 9761, whose string of implications is:
    203 \(\Rightarrow\) 369

The conclusion Form 209 \( \not \Rightarrow \) Form 203 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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