Implications

This non-implication, Form 152 \( \not \Rightarrow \) Form 99, 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>

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

The conclusion Form 152 \( \not \Rightarrow \) Form 99 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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