Implications

This non-implication, Form 167 \( \not \Rightarrow \) Form 322, 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: 736, Form 379 \( \not \Rightarrow \) Form 355 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 355 <p> \(KW(\aleph_0,\infty)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of sets: For every denumerable set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • 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

The conclusion Form 167 \( \not \Rightarrow \) Form 322 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

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