Implications

This non-implication, Form 94 \( \not \Rightarrow \) Form 213, 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: 2650, whose string of implications is:
    40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 94
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1376, Form 40 \( \not \Rightarrow \) Form 349 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 349 <p> \(MC(\infty,\aleph_0)\): For every set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6344, whose string of implications is:
    213 \(\Rightarrow\) 85 \(\Rightarrow\) 349

The conclusion Form 94 \( \not \Rightarrow \) Form 213 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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