Implications

This non-implication, Form 163 \( \not \Rightarrow \) Form 359, 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: 10962, whose string of implications is:
    163 \(\Rightarrow\) 163
  • 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: 1802, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18

The conclusion Form 163 \( \not \Rightarrow \) Form 359 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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