Implications

This non-implication, Form 317 \( \not \Rightarrow \) Form 328, whose code is 6, 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 5. In this case, it's Code 3: 621, Form 317 \( \not \Rightarrow \) Form 84 whose summary information is:
    Hypothesis Statement
    Form 317 <p> <strong>Weak Sikorski Theorem:</strong>  If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).

    Conclusion Statement
    Form 84 <p> \(E(II,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7551, whose string of implications is:
    328 \(\Rightarrow\) 126 \(\Rightarrow\) 82 \(\Rightarrow\) 84

The conclusion Form 317 \( \not \Rightarrow \) Form 328 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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