Implications

This non-implication, Form 216 \( \not \Rightarrow \) Form 99, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2002, whose string of implications is:
    49 \(\Rightarrow\) 30 \(\Rightarrow\) 10 \(\Rightarrow\) 216
  • 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 216 \( \not \Rightarrow \) Form 99 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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