Implications

This non-implication, Form 295 \( \not \Rightarrow \) Form 89, whose code is 4, 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 3. In this case, it's Code 3: 980, Form 295 \( \not \Rightarrow \) Form 118 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </p>

    Conclusion Statement
    Form 118 <p> Every linearly orderable topological space is normal.  <a href="/books/28">Birkhoff [1967]</a>, p 241. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6386, whose string of implications is:
    89 \(\Rightarrow\) 90 \(\Rightarrow\) 118

The conclusion Form 295 \( \not \Rightarrow \) Form 89 then follows.

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

Name Statement
\(\cal M29\) Pincus' Model II Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)
\(\cal M44\) Pincus' Model VI This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((B)\)
\(\cal N38\) Howard/Rubin Model I Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)
\(\cal N48\) Pincus' Model XI \(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\)

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