Implications

This non-implication, Form 102 \( \not \Rightarrow \) 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: 4204, whose string of implications is:
    60 \(\Rightarrow\) 62 \(\Rightarrow\) 102
  • 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 102 \( \not \Rightarrow \) 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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