Implications

This non-implication, Form 75 \( \not \Rightarrow \) Form 67, 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: 90, whose string of implications is:
    3 \(\Rightarrow\) 4 \(\Rightarrow\) 405 \(\Rightarrow\) 75
  • A proven non-implication whose code is 3. In this case, it's Code 3: 235, Form 3 \( \not \Rightarrow \) Form 5 whose summary information is:
    Hypothesis Statement
    Form 3  \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

    Conclusion Statement
    Form 5 <p> \(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5628, whose string of implications is:
    67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 5

The conclusion Form 75 \( \not \Rightarrow \) Form 67 then follows.

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

Name Statement
\(\cal M6\) Sageev's Model I Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)

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