Implications

This non-implication, Form 307 \( \not \Rightarrow \) Form 309, 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: 945, Form 307 \( \not \Rightarrow \) Form 93 whose summary information is:
    Hypothesis Statement
    Form 307 <p> If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\). </p>

    Conclusion Statement
    Form 93 <p> There is a non-measurable subset of \({\Bbb R}\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9924, whose string of implications is:
    309 \(\Rightarrow\) 93

The conclusion Form 307 \( \not \Rightarrow \) Form 309 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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