Implications

This non-implication, Form 80 \( \not \Rightarrow \) Form 133, 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: 10180, whose string of implications is:
    122 \(\Rightarrow\) 80
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1078, Form 122 \( \not \Rightarrow \) Form 151 whose summary information is:
    Hypothesis Statement
    Form 122 <p> \(C(WO,<\aleph_{0})\): Every well ordered 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: 7596, whose string of implications is:
    133 \(\Rightarrow\) 231 \(\Rightarrow\) 151

The conclusion Form 80 \( \not \Rightarrow \) Form 133 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 \((C)\)
\(\cal N15\) Brunner/Howard Model I \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\)

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