Implications

This non-implication, Form 371 \( \not \Rightarrow \) Form 295, 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: 920, Form 371 \( \not \Rightarrow \) Form 64 whose summary information is:
    Hypothesis Statement
    Form 371 <p> There is an infinite, compact, Hausdorff, extremally disconnected topological space.  <a href="/excerpts/Morillon-1993-1">Morillon [1993]</a>. </p>

    Conclusion Statement
    Form 64 <p> \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2063, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 64

The conclusion Form 371 \( \not \Rightarrow \) Form 295 then follows.

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

Name Statement
\(\cal M37\) Monro's Model III This is a generic extension of <a href="/models/Cohen-1">\(\cal M1\)</a> in which there is an amorphous set (<a href="/form-classes/howard-rubin-64">Form 64</a> is false) and \(C(\infty,2)\) (<a href="/form-classes/howard-rubin-88">Form 88</a>) is false
\(\cal N1\) 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 \(A\)
\(\cal N43\) Brunner's Model II The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint

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