Implications

This non-implication, Form 382 \( \not \Rightarrow \) Form 45-n, whose code is 6, 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: 5831, whose string of implications is:
    333 \(\Rightarrow\) 67 \(\Rightarrow\) 381 \(\Rightarrow\) 382
  • A proven non-implication whose code is 5. In this case, it's Code 3: 665, Form 333 \( \not \Rightarrow \) Form 288-n whose summary information is:
    Hypothesis Statement
    Form 333 <p> \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  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 \(x\) and \(|f(x)|\) is odd. </p>

    Conclusion Statement
    Form 288-n <p> If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2467, whose string of implications is:
    45-n \(\Rightarrow\) 33-n \(\Rightarrow\) 47-n \(\Rightarrow\) 288-n

The conclusion Form 382 \( \not \Rightarrow \) Form 45-n then follows.

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

Name Statement
\(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)

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