Implications

This non-implication, Form 334 \( \not \Rightarrow \) Form 168, whose code is 6, 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 5. In this case, it's Code 3: 677, Form 334 \( \not \Rightarrow \) Form 388 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 388 <p> Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7232, whose string of implications is:
    168 \(\Rightarrow\) 100 \(\Rightarrow\) 347 \(\Rightarrow\) 40 \(\Rightarrow\) 43 \(\Rightarrow\) 388

The conclusion Form 334 \( \not \Rightarrow \) Form 168 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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