Implications

This non-implication, Form 163 \( \not \Rightarrow \) Form 295, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 25, Form 163 \( \not \Rightarrow \) Form 198 whose summary information is:
    Hypothesis Statement
    Form 163 <p> Every non-well-orderable set has an infinite, Dedekind finite subset. </p>

    Conclusion Statement
    Form 198 <p> For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2012, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 198

The conclusion Form 163 \( \not \Rightarrow \) Form 295 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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