Implications

This non-implication, Form 128 \( \not \Rightarrow \) Form 333, 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: 2655, whose string of implications is:
    40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 128
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6061, whose string of implications is:
    333 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 203

The conclusion Form 128 \( \not \Rightarrow \) Form 333 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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