Implications

This non-implication, Form 355 \( \not \Rightarrow \) Form 126, 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: 10005, whose string of implications is:
    322 \(\Rightarrow\) 355
  • A proven non-implication whose code is 5. In this case, it's Code 3: 652, Form 322 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 322 <p> \(KW(WO,\infty)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of sets:</strong> For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See <a href="/form-classes/howard-rubin-15">Form 15</a>). </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: 6251, whose string of implications is:
    126 \(\Rightarrow\) 82 \(\Rightarrow\) 83 \(\Rightarrow\) 64 \(\Rightarrow\) 390

The conclusion Form 355 \( \not \Rightarrow \) Form 126 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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