Implications

This non-implication, Form 163 \( \not \Rightarrow \) Form 398, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1317, Form 163 \( \not \Rightarrow \) Form 324 whose summary information is:
    Hypothesis Statement
    Form 163 <p> Every non-well-orderable set has an infinite, Dedekind finite subset. </p>

    Conclusion Statement
    Form 324 <p> \(KW(WO,WO)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets:</strong> For every well ordered set \(M\) of well orderable sets, 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8936, whose string of implications is:
    398 \(\Rightarrow\) 322 \(\Rightarrow\) 324

The conclusion Form 163 \( \not \Rightarrow \) Form 398 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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