Implications

This non-implication, Form 135 \( \not \Rightarrow \) Form 193, whose code is 6, 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 5. In this case, it's Code 3: 400, Form 135 \( \not \Rightarrow \) Form 390 whose summary information is:
    Hypothesis Statement
    Form 135 <p> If \(X\) is a \(T_2\) space with at least two points and \(X^{Y}\) is hereditarily metacompact then \(Y\) is  countable. (A space is <em>metacompact</em> if every open cover has an open point finite refinement. If \(B\) and \(B'\) are covers of a space \(X\), then \(B'\) is a <em>refinement</em> of \(B\) if \((\forall x\in B')(\exists y\in B)(x\subseteq y)\). \(B\) is <em>point finite</em> if \((\forall t\in X)\) there are only finitely many \(x\in B\) such that \(t\in x\).) <a href="/excerpts/van-Douwen-1980">van Douwen [1980]</a> </p>

    Conclusion Statement
    Form 390 <p> Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. <a href="/excerpts/Ash-1981-1">Ash [1983]</a>. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7679, whose string of implications is:
    193 \(\Rightarrow\) 188 \(\Rightarrow\) 106 \(\Rightarrow\) 126 \(\Rightarrow\) 82 \(\Rightarrow\) 83 \(\Rightarrow\) 64 \(\Rightarrow\) 390

The conclusion Form 135 \( \not \Rightarrow \) Form 193 then follows.

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

Name Statement
\(\cal N1\) 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 \(A\)

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