Implications

This non-implication, Form 80 \( \not \Rightarrow \) Form 274, 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: 2424, whose string of implications is:
    165 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1194, Form 165 \( \not \Rightarrow \) Form 274 whose summary information is:
    Hypothesis Statement
    Form 165 <p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function. </p>

    Conclusion Statement
    Form 274 <p> There is a cardinal number \(x\) and an \(n\in\omega\) such that \(\neg(x\) adj\(^n\, x^2)\). (The expression ``\(x\) adj\(^n\, ya\)" means there are cardinals \(z_0,\ldots, z_n\) such that \(z_0 = x\) and \(z_n = y\) and for all \(i,\ 0\le i < n,\ z_i< z_{i+1}\) and if  \(z_i < z\le z_{i+1}\), then \(z = z_{i+1}.)\) (Compare with <a href="/form-class-members/howard-rubin-0-a">[0 A]</a>). </p>

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

The conclusion Form 80 \( \not \Rightarrow \) Form 274 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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