Implications

This non-implication, Form 118 \( \not \Rightarrow \) Form 346, 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: 7563, whose string of implications is:
    346 \(\Rightarrow\) 126 \(\Rightarrow\) 131

The conclusion Form 118 \( \not \Rightarrow \) Form 346 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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