Implications

This non-implication, Form 34 \( \not \Rightarrow \) Form 395, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6133, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 94 \(\Rightarrow\) 34
  • A proven non-implication whose code is 5. In this case, it's Code 3: 234, Form 91 \( \not \Rightarrow \) Form 330 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </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: 9536, whose string of implications is:
    395 \(\Rightarrow\) 396 \(\Rightarrow\) 330

The conclusion Form 34 \( \not \Rightarrow \) Form 395 then follows.

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

Name Statement
\(\cal N15\) Brunner/Howard Model I \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\)
\(\cal N41\) Another variation of \(\cal N3\) \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)

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