Implications

This non-implication, Form 178-n-N \( \not \Rightarrow \) Form 59-le, 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: 2126, whose string of implications is:
    317 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 178-n-N
  • A proven non-implication whose code is 5. In this case, it's Code 3: 619, Form 317 \( \not \Rightarrow \) Form 59-le 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 59-le <p> If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\).<br /> <a href="/articles/Mathias-1979">Mathias [1979]</a>, p 120. </p>

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

The conclusion Form 178-n-N \( \not \Rightarrow \) Form 59-le then follows.

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

Name Statement
\(\cal N19(\precsim)\) Tsukada's Model Let \((P,\precsim)\) be a partiallyordered set that is not well ordered; Let \(Q\) be a countably infinite set,disjoint from \(P\); and let \(I=P\cup Q\)

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