Implications

This non-implication, Form 352 \( \not \Rightarrow \) 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: 9687, whose string of implications is:
    16 \(\Rightarrow\) 352
  • A proven non-implication whose code is 5. In this case, it's Code 3: 31, Form 16 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 16 <p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function. </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 352 \( \not \Rightarrow \) 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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