Implications

This non-implication, Form 34 \( \not \Rightarrow \) Form 334, 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: 692, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 34
  • 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: 5770, whose string of implications is:
    334 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 305 \(\Rightarrow\) 307

The conclusion Form 34 \( \not \Rightarrow \) Form 334 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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