Implications

This non-implication, Form 189 \( \not \Rightarrow \) Form 174-alpha, 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: 9615, whose string of implications is:
    191 \(\Rightarrow\) 189
  • A proven non-implication whose code is 5. In this case, it's Code 3: 521, Form 191 \( \not \Rightarrow \) Form 357 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 357 <p> \(KW(\aleph_0,\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3664, whose string of implications is:
    174-alpha \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 357

The conclusion Form 189 \( \not \Rightarrow \) Form 174-alpha 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 \(\le_n\)

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