Implications

This non-implication, Form 170 \( \not \Rightarrow \) Form 359, 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: 1421, Form 170 \( \not \Rightarrow \) Form 369 whose summary information is:
    Hypothesis Statement
    Form 170 <p> \(\aleph_{1}\le 2^{\aleph_{0}}\). </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: 7087, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 101 \(\Rightarrow\) 100 \(\Rightarrow\) 369

The conclusion Form 170 \( \not \Rightarrow \) Form 359 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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