Implications

This non-implication, Form 151 \( \not \Rightarrow \) Form 149, 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: 3494, whose string of implications is:
    40 \(\Rightarrow\) 231 \(\Rightarrow\) 151
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1331, Form 40 \( \not \Rightarrow \) Form 329 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 329 <p> \(MC(\infty,WO)\): For  every set \(M\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5158, whose string of implications is:
    149 \(\Rightarrow\) 67 \(\Rightarrow\) 329

The conclusion Form 151 \( \not \Rightarrow \) Form 149 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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