Implications

This non-implication, Form 167 \( \not \Rightarrow \) Form 378, 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: 9946, whose string of implications is:
    379 \(\Rightarrow\) 167
  • A proven non-implication whose code is 5. In this case, it's Code 3: 737, Form 379 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 379 <p> \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). </p>

    Conclusion Statement
    Form 390 <p> Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. <a href="/excerpts/Ash-1981-1">Ash [1983]</a>. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4878, whose string of implications is:
    378 \(\Rightarrow\) 11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n \(\Rightarrow\) 64 \(\Rightarrow\) 390

The conclusion Form 167 \( \not \Rightarrow \) Form 378 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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