Implications

This non-implication, Form 357 \( \not \Rightarrow \) Form 75, 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: 8939, whose string of implications is:
    322 \(\Rightarrow\) 324 \(\Rightarrow\) 357
  • 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: 5907, whose string of implications is:
    75 \(\Rightarrow\) 404 \(\Rightarrow\) 390

The conclusion Form 357 \( \not \Rightarrow \) Form 75 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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