Implications

This non-implication, Form 10 \( \not \Rightarrow \) Form 305, 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: 502, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 10
  • A proven non-implication whose code is 3. In this case, it's Code 3: 166, Form 43 \( \not \Rightarrow \) Form 307 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 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9956, whose string of implications is:
    305 \(\Rightarrow\) 307

The conclusion Form 10 \( \not \Rightarrow \) Form 305 then follows.

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

Name Statement
\(\cal M39(\kappa,\lambda)\) Kanovei's model II This model depends on the two cardinals \(\kappa < \lambda\) such that both \(\kappa\) and \(\lambda\) have cofinality \(>\omega\) and neither \(\kappa\) nor \(\lambda\) can be written as \(\theta^+\) where \(\theta\) is a cardinal of countable cofinality and such that \(\aleph_2 \le\kappa\)

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