Implications

This non-implication, Form 375 \( \not \Rightarrow \) Form 202, 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: 10245, whose string of implications is:
    43 \(\Rightarrow\) 375
  • A proven non-implication whose code is 3. In this case, it's Code 3: 124, Form 43 \( \not \Rightarrow \) Form 39 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 39 <p> \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p. 202. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3264, whose string of implications is:
    202 \(\Rightarrow\) 40 \(\Rightarrow\) 39

The conclusion Form 375 \( \not \Rightarrow \) Form 202 then follows.

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

Name Statement
\(\cal M1(\langle\omega_1\rangle)\) Cohen/Pincus Model Pincus extends the methods of Cohen and adds a generic \(\omega_1\)-sequence, \(\langle I_{\alpha}: \alpha\in\omega_1\rangle\), of denumerable sets, where \(I_0\) is a denumerable set of generic reals, each \(I_{\alpha+1}\) is a generic set of enumerations of \(I_{\alpha}\), and for a limit ordinal \(\lambda\),\(I_{\lambda}\) is a generic set of choice functions for \(\{I_{\alpha}:\alpha \le \lambda\}\)

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