Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 395, whose code is 6, 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: 11016, whose string of implications is:
    191 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 519, Form 191 \( \not \Rightarrow \) Form 350 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 350 <p> \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9537, whose string of implications is:
    395 \(\Rightarrow\) 396 \(\Rightarrow\) 330 \(\Rightarrow\) 350

The conclusion Form 0 \( \not \Rightarrow \) Form 395 then follows.

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

Name Statement
\(\cal N41\) 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 \(\le_n\)

Edit | Back