Implications

This non-implication, Form 24 \( \not \Rightarrow \) Form 367, 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: 358, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 24
  • A proven non-implication whose code is 3. In this case, it's Code 3: 139, Form 43 \( \not \Rightarrow \) Form 93 whose summary information is:
    Hypothesis Statement
    Form 43 <p> \(DC(\omega)\) (DC), <strong>Principle of Dependent Choices:</strong> If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See <a href="/articles/Tarski-1948">Tarski [1948]</a>, p 96, <a href="/articles/Levy-1964">Levy [1964]</a>, p. 136. </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: 9368, whose string of implications is:
    367 \(\Rightarrow\) 366 \(\Rightarrow\) 93

The conclusion Form 24 \( \not \Rightarrow \) Form 367 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 \(\Bbb R\)
\(\cal M38\) Shelah's Model II In a model of \(ZFC +\) "\(\kappa\) is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse \(\kappa\) to \(\aleph_1\) similarly to <a href="/articles/Solovay-1970">Solovay [1970]</a>

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