Implications

This non-implication, Form 97 \( \not \Rightarrow \) Form 60, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 935, Form 97 \( \not \Rightarrow \) Form 85 whose summary information is:
    Hypothesis Statement
    Form 97 <p> <strong>Cardinal Representatives:</strong> For every set \(A\) there is a function \(c\) with domain \({\cal P}(A)\) such that for all \(x, y\in {\cal P}(A)\), (i) \(c(x) = c(y) \leftrightarrow x\approx y\) and (ii) \(c(x)\approx x\).  <a href="/books/8">Jech [1973b]</a> p 154. </p>

    Conclusion Statement
    Form 85 <p> \(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  <a href="/books/8">Jech [1973b]</a> p 115 prob 7.13. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10183, whose string of implications is:
    60 \(\Rightarrow\) 85

The conclusion Form 97 \( \not \Rightarrow \) Form 60 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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