Implications

This non-implication, Form 37 \( \not \Rightarrow \) Form 352, 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: 5564, whose string of implications is:
    334 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 37
  • A proven non-implication whose code is 5. In this case, it's Code 3: 667, Form 334 \( \not \Rightarrow \) Form 18 whose summary information is:
    Hypothesis Statement
    Form 334 <p> \(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 2\), 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 even. </p>

    Conclusion Statement
    Form 18 <p> \(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2330, whose string of implications is:
    352 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18

The conclusion Form 37 \( \not \Rightarrow \) Form 352 then follows.

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

Name Statement
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).

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