Implications

This non-implication, Form 27 \( \not \Rightarrow \) Form 15, 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: 2649, whose string of implications is:
    40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 27
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1392, Form 40 \( \not \Rightarrow \) Form 356 whose summary information is:
    Hypothesis Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </p>

    Conclusion Statement
    Form 356 <p>  \(KW(\infty,\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a family of denumerable sets: For every 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: 1699, whose string of implications is:
    15 \(\Rightarrow\) 323 \(\Rightarrow\) 356

The conclusion Form 27 \( \not \Rightarrow \) Form 15 then follows.

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

Name Statement
\(\cal N33\) Howard/H\.Rubin/J\.Rubin Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all boundedsubsets of \(A\)

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