Implications

This non-implication, Form 376 \( \not \Rightarrow \) Form 381, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10019, whose string of implications is:
    9 \(\Rightarrow\) 376
  • A proven non-implication whose code is 3. In this case, it's Code 3: 114, Form 9 \( \not \Rightarrow \) Form 382 whose summary information is:
    Hypothesis Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

    Conclusion Statement
    Form 382 <p> <strong>DUMN</strong>:  The disjoint union of metrizable spaces is normal. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10026, whose string of implications is:
    381 \(\Rightarrow\) 382

The conclusion Form 376 \( \not \Rightarrow \) Form 381 then follows.

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

Name Statement
\(\cal N53\) Good/Tree/Watson Model I Let \(A=\bigcup \{Q_n:\ n\in\omega\}\), where \(Q_n=\{a_{n,q}:q\in \Bbb{Q}\}\)
\(\cal N57\) The set of atoms \(A=\cup\{A_{n}:n\in\aleph_{1}\}\), where\(A_{n}=\{a_{nx}:x\in B(0,1)\}\) and \(B(0,1)\) is the set of points on theunit circle centered at 0 The group of permutations \(\cal{G}\) is thegroup of all permutations on \(A\) which rotate the \(A_{n}\)'s by an angle\(\theta_{n}\in\Bbb{R}\) and supports are countable

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