Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 91, 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: 258, Form 204 \( \not \Rightarrow \) Form 203 whose summary information is:
    Hypothesis Statement
    Form 204 <p> For every infinite \(X\), there is a function from \(X\) onto \(2X\). </p>

    Conclusion Statement
    Form 203 <p> \(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5984, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 203

The conclusion Form 204 \( \not \Rightarrow \) Form 91 then follows.

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

Name Statement
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them
\(\cal M6\) 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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