Implications

This non-implication, Form 108 \( \not \Rightarrow \) Form 284, 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: 615, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 16 \(\Rightarrow\) 6 \(\Rightarrow\) 5 \(\Rightarrow\) 38 \(\Rightarrow\) 108
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1210, Form 43 \( \not \Rightarrow \) Form 280 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 280 <p> There is a complete separable metric space with a subset which does not have the Baire property. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4355, whose string of implications is:
    284 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 142 \(\Rightarrow\) 280

The conclusion Form 108 \( \not \Rightarrow \) Form 284 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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