Implications

This non-implication, Form 308-p \( \not \Rightarrow \) Form 23, 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: 9766, whose string of implications is:
    62 \(\Rightarrow\) 308-p
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1075, Form 62 \( \not \Rightarrow \) Form 151 whose summary information is:
    Hypothesis Statement
    Form 62 <p> \(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function. </p>

    Conclusion Statement
    Form 151 <p> \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see <a href="/notes/note-27">note 27</a> for \(UT(WO,\kappa,WO)\).) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10206, whose string of implications is:
    23 \(\Rightarrow\) 151

The conclusion Form 308-p \( \not \Rightarrow \) Form 23 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 \(\{I_{\alpha}:\alpha \le \lambda\}\)
\(\cal M29\) Pincus' Model II Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)
\(\cal M43\) Pincus' Model V This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((A)\)
\(\cal M44\) Pincus' Model VI This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((B)\)
\(\cal M45\) Pincus' Model VII This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\)

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