Implications

This non-implication, Form 135 \( \not \Rightarrow \) Form 400, 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: 387, Form 135 \( \not \Rightarrow \) Form 111 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 111 <p> \(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9553, whose string of implications is:
    400 \(\Rightarrow\) 401 \(\Rightarrow\) 327 \(\Rightarrow\) 250 \(\Rightarrow\) 111

The conclusion Form 135 \( \not \Rightarrow \) Form 400 then follows.

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

Name Statement
\(\cal N2(\aleph_{\alpha})\) Jech's Model This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)

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