Implications

This non-implication, Form 92 \( \not \Rightarrow \) Form 391, 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: 3513, whose string of implications is:
    40 \(\Rightarrow\) 337 \(\Rightarrow\) 92
  • 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: 9484, whose string of implications is:
    391 \(\Rightarrow\) 399 \(\Rightarrow\) 323 \(\Rightarrow\) 356

The conclusion Form 92 \( \not \Rightarrow \) Form 391 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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