Implications

This non-implication, Form 308-p \( \not \Rightarrow \) Form 286, 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: 2064, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 308-p
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1385, Form 295 \( \not \Rightarrow \) Form 350 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </p>

    Conclusion Statement
    Form 350 <p> \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2962, whose string of implications is:
    286 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 350

The conclusion Form 308-p \( \not \Rightarrow \) Form 286 then follows.

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

Name Statement
\(\cal M29\) 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 \(I_n\)

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