Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 292, 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: 6727, whose string of implications is:
    292 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 203

The conclusion Form 204 \( \not \Rightarrow \) Form 292 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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