Implications

This non-implication, Form 223 \( \not \Rightarrow \) Form 340, 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: 5900, whose string of implications is:
    99 \(\Rightarrow\) 70 \(\Rightarrow\) 206 \(\Rightarrow\) 223
  • A proven non-implication whose code is 5. In this case, it's Code 3: 245, Form 99 \( \not \Rightarrow \) Form 80 whose summary information is:
    Hypothesis 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>

    Conclusion Statement
    Form 80 <p> \(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9353, whose string of implications is:
    340 \(\Rightarrow\) 341 \(\Rightarrow\) 10 \(\Rightarrow\) 80

The conclusion Form 223 \( \not \Rightarrow \) Form 340 then follows.

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

Name Statement
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).

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