Implications

This non-implication, Form 308-p \( \not \Rightarrow \) Form 164, 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: 4203, whose string of implications is:
    60 \(\Rightarrow\) 62 \(\Rightarrow\) 308-p
  • A proven non-implication whose code is 3. In this case, it's Code 3: 147, Form 60 \( \not \Rightarrow \) Form 79 whose summary information is:
    Hypothesis Statement
    Form 60 <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </p>

    Conclusion Statement
    Form 79 <p> \({\Bbb R}\) can be well ordered.  <a href="/articles/hilbert-1900">Hilbert [1900]</a>, p 263. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6808, whose string of implications is:
    164 \(\Rightarrow\) 91 \(\Rightarrow\) 79

The conclusion Form 308-p \( \not \Rightarrow \) Form 164 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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