Implications

This non-implication, Form 190 \( \not \Rightarrow \) Form 359, 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: 7695, whose string of implications is:
    191 \(\Rightarrow\) 189 \(\Rightarrow\) 190
  • 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: 1792, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 357

The conclusion Form 190 \( \not \Rightarrow \) Form 359 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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