Implications

This non-implication, Form 329 \( \not \Rightarrow \) Form 258, 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: 9974, whose string of implications is:
    60 \(\Rightarrow\) 329
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1308, Form 60 \( \not \Rightarrow \) Form 322 whose summary information is:
    Hypothesis Statement
    Form 60 <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </p>

    Conclusion Statement
    Form 322 <p> \(KW(WO,\infty)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of sets:</strong> For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See <a href="/form-classes/howard-rubin-15">Form 15</a>). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8354, whose string of implications is:
    258 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 322

The conclusion Form 329 \( \not \Rightarrow \) Form 258 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered 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 finitesubsets of \(A\)

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