Implications

This non-implication, Form 48-K \( \not \Rightarrow \) Form 427, 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: 2229, whose string of implications is:
    317 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 121 \(\Rightarrow\) 122 \(\Rightarrow\) 48-K
  • A proven non-implication whose code is 5. In this case, it's Code 3: 625, Form 317 \( \not \Rightarrow \) Form 125 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 125 <p> There does not exist an infinite, compact connected \(p\) space. (A \(p\) <em>space</em> is a \(T_2\) space in which the intersection of any well orderable family of open sets is open.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5751, whose string of implications is:
    427 \(\Rightarrow\) 67 \(\Rightarrow\) 144 \(\Rightarrow\) 125

The conclusion Form 48-K \( \not \Rightarrow \) Form 427 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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