Implications

This non-implication, Form 216 \( \not \Rightarrow \) Form 129, 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: 10311, whose string of implications is:
    217 \(\Rightarrow\) 216
  • A proven non-implication whose code is 3. In this case, it's Code 3: 204, Form 217 \( \not \Rightarrow \) Form 9 whose summary information is:
    Hypothesis Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

    Conclusion Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10080, whose string of implications is:
    129 \(\Rightarrow\) 9

The conclusion Form 216 \( \not \Rightarrow \) Form 129 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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