Implications

This non-implication, Form 9 \( \not \Rightarrow \) Form 310, 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: 345, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 9
  • A proven non-implication whose code is 3. In this case, it's Code 3: 238, Form 43 \( \not \Rightarrow \) Form 142 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 142 <p> \(\neg  PB\):  There is a set of reals without the property of Baire.  <a href="/books/8">Jech [1973b]</a>, p. 7. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9891, whose string of implications is:
    310 \(\Rightarrow\) 142

The conclusion Form 9 \( \not \Rightarrow \) Form 310 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 \(\Bbb R\)
\(\cal M18\) Shelah's Model I Shelah modified Solovay's model, <a href="/models/Solovay-1">\(\cal M5\)</a>, and constructed a model without using an inaccessible cardinal in which the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true and every set of reals has the property of Baire (<a href="/form-classes/howard-rubin-142">Form142</a> is false)

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