Implications

This non-implication, Form 207-alpha \( \not \Rightarrow \) Form 59-le, 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: 10310, whose string of implications is:
    23 \(\Rightarrow\) 207-alpha
  • A proven non-implication whose code is 5. In this case, it's Code 3: 38, Form 23 \( \not \Rightarrow \) Form 59-le whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </p>

    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 207-alpha \( \not \Rightarrow \) 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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