Implications

This non-implication, Form 135 \( \not \Rightarrow \) Form 335-n, 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: 9219, whose string of implications is:
    335-n \(\Rightarrow\) 333 \(\Rightarrow\) 67 \(\Rightarrow\) 126 \(\Rightarrow\) 82 \(\Rightarrow\) 83 \(\Rightarrow\) 64 \(\Rightarrow\) 390

The conclusion Form 135 \( \not \Rightarrow \) Form 335-n 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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