Implications

This non-implication, Form 154 \( \not \Rightarrow \) Form 407, 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: 9627, whose string of implications is:
    43 \(\Rightarrow\) 154
  • A proven non-implication whose code is 3. In this case, it's Code 3: 68, Form 43 \( \not \Rightarrow \) Form 410 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 410 <p> <strong>RC (Reflexive Compactness)</strong>: The closed unit ball of a reflexive normed space is compact for the weak topology. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1309, whose string of implications is:
    407 \(\Rightarrow\) 14 \(\Rightarrow\) 410

The conclusion Form 154 \( \not \Rightarrow \) Form 407 then follows.

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

Name Statement
\(\cal M27\) Pincus/Solovay Model I Let \(\cal M_1\) be a model of \(ZFC + V =L\)

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