Implications

This non-implication, Form 194 \( \not \Rightarrow \) Form 302, 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: 592, whose string of implications is:
    8 \(\Rightarrow\) 16 \(\Rightarrow\) 194
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8927, whose string of implications is:
    302 \(\Rightarrow\) 392 \(\Rightarrow\) 393 \(\Rightarrow\) 165 \(\Rightarrow\) 330

The conclusion Form 194 \( \not \Rightarrow \) Form 302 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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