Implications

This non-implication, Form 128 \( \not \Rightarrow \) Form 330, 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: 409, whose string of implications is:
    8 \(\Rightarrow\) 9 \(\Rightarrow\) 128
  • A proven non-implication whose code is 3. In this case, it's Code 3: 282, Form 8 \( \not \Rightarrow \) Form 330 whose summary information is:
    Hypothesis Statement
    Form 8 <p> \(C(\aleph_{0},\infty)\): </p>

    Conclusion Statement
    Form 330 <p> \(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

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

The conclusion Form 128 \( \not \Rightarrow \) Form 330 then follows.

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

Name Statement
\(\cal N15\) Brunner/Howard Model I \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\)

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