Implications

This non-implication, Form 369 \( \not \Rightarrow \) Form 117, 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: 1305, Form 369 \( \not \Rightarrow \) Form 318 whose summary information is:
    Hypothesis Statement
    Form 369 <p> If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). </p>

    Conclusion Statement
    Form 318 <p> \(\aleph_1\) is not measurable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7447, whose string of implications is:
    117 \(\Rightarrow\) 319 \(\Rightarrow\) 320 \(\Rightarrow\) 318

The conclusion Form 369 \( \not \Rightarrow \) Form 117 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M5(\aleph)\) Solovay's Model An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)

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