Implications

This non-implication, Form 223 \( \not \Rightarrow \) Form 356, 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: 4862, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 206 \(\Rightarrow\) 223
  • A proven non-implication whose code is 5. In this case, it's Code 3: 134, Form 63 \( \not \Rightarrow \) Form 356 whose summary information is:
    Hypothesis Statement
    Form 63 <p> \(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. <br /> <a href="/books/8">Jech [1973b]</a>, p 172 prob 8.5. </p>

    Conclusion Statement
    Form 356 <p>  \(KW(\infty,\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a family of denumerable sets: For every set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\)  then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 223 \( \not \Rightarrow \) Form 356 then follows.

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

Name Statement
\(\cal N33\) Howard/H\.Rubin/J\.Rubin 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 boundedsubsets of \(A\)
\(\cal N41\) Another variation of \(\cal N3\) \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)

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