Implications

This non-implication, Form 152 \( \not \Rightarrow \) Form 384, 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: 10151, whose string of implications is:
    214 \(\Rightarrow\) 152
  • A proven non-implication whose code is 3. In this case, it's Code 3: 269, Form 214 \( \not \Rightarrow \) Form 99 whose summary information is:
    Hypothesis Statement
    Form 214 <p> \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). </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: 1322, whose string of implications is:
    384 \(\Rightarrow\) 14 \(\Rightarrow\) 99

The conclusion Form 152 \( \not \Rightarrow \) Form 384 then follows.

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

Name Statement
\(\cal M43\) Pincus' Model V This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((A)\)
\(\cal M45\) Pincus' Model VII This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\)

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