Implications

This non-implication, Form 118 \( \not \Rightarrow \) Form 126, 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: 1037, Form 118 \( \not \Rightarrow \) Form 131 whose summary information is:
    Hypothesis Statement
    Form 118 <p> Every linearly orderable topological space is normal.  <a href="/books/28">Birkhoff [1967]</a>, p 241. </p>

    Conclusion Statement
    Form 131 <p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9936, whose string of implications is:
    126 \(\Rightarrow\) 131

The conclusion Form 118 \( \not \Rightarrow \) Form 126 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)
\(\cal N24(n)\) An extension of \(\cal N24\) to \(n\)-element sets, \(n>1\).\(A=\bigcup B\), where \( B=\{b_i: i\in\omega\}\) is a pairwise disjoint setof \(n\)-element sets \(\cal G\) is the group of all permutations of \(A\)which are permutations of \(B\); and \(S\) is the set of all finite subsets of\(A\)
\(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\) The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\)

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