Implications

This non-implication, Form 326 \( \not \Rightarrow \) Form 317, whose code is 6, 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: 9897, whose string of implications is:
    49 \(\Rightarrow\) 326
  • A proven non-implication whose code is 5. In this case, it's Code 3: 119, Form 49 \( \not \Rightarrow \) Form 99 whose summary information is:
    Hypothesis Statement
    Form 49 <p> <strong>Order Extension Principle:</strong> Every partial ordering can be extended to a linear ordering.  <a href="/articles/Tarski-1924">Tarski [1924]</a>, p 78. </p>

    Conclusion Statement
    Form 99 <p> <strong>Rado's Selection Lemma:</strong> Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1346, whose string of implications is:
    317 \(\Rightarrow\) 14 \(\Rightarrow\) 99

The conclusion Form 326 \( \not \Rightarrow \) Form 317 then follows.

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

Name Statement
\(\cal N52\) Felgner/Truss Model Let \((\cal B,\prec)\) be a countableuniversal homogeneous linearly ordered Boolean algebra, (i.e., \(<\) is alinear ordering extending the Boolean partial ordering on \(B\))

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