Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 376, 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: 10961, whose string of implications is:
    163 \(\Rightarrow\) 0
  • A proven non-implication whose code is 3. In this case, it's Code 3: 871, Form 163 \( \not \Rightarrow \) Form 18 whose summary information is:
    Hypothesis Statement
    Form 163 <p> Every non-well-orderable set has an infinite, Dedekind finite subset. </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: 7663, whose string of implications is:
    376 \(\Rightarrow\) 167 \(\Rightarrow\) 18

The conclusion Form 0 \( \not \Rightarrow \) Form 376 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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