Implications

This non-implication, Form 279 \( \not \Rightarrow \) Form 203, 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: 3332, whose string of implications is:
    40 \(\Rightarrow\) 43 \(\Rightarrow\) 279
  • A proven non-implication whose code is 3. In this case, it's Code 3: 257, Form 40 \( \not \Rightarrow \) Form 203 whose summary information is:
    Hypothesis Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 279 \( \not \Rightarrow \) Form 203 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

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