Implications

This non-implication, Form 307 \( \not \Rightarrow \) Form 21, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5768, whose string of implications is:
    333 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 305 \(\Rightarrow\) 307
  • A proven non-implication whose code is 5. In this case, it's Code 3: 664, Form 333 \( \not \Rightarrow \) Form 250 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 250 <p> \((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family 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: 7507, whose string of implications is:
    21 \(\Rightarrow\) 23 \(\Rightarrow\) 151 \(\Rightarrow\) 122 \(\Rightarrow\) 250

The conclusion Form 307 \( \not \Rightarrow \) Form 21 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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