Implications

This non-implication, Form 132 \( \not \Rightarrow \) Form 341, 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: 1020, whose string of implications is:
    9 \(\Rightarrow\) 17 \(\Rightarrow\) 132
  • A proven non-implication whose code is 3. In this case, it's Code 3: 174, Form 9 \( \not \Rightarrow \) Form 341 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 341 <p> Every Lindel&ouml;f metric space is second countable. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 132 \( \not \Rightarrow \) Form 341 then follows.

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

Name Statement
\(\cal N58\) Keremedis/Tachtsis Model 2: For each \(n\in\omega-\{0\}\), let\(A_n=\{({i\over n}) (\cos t,\sin t): t\in [0.2\pi)\}\) and let the set of atoms\(A=\bigcup \{A_n: n\in\omega-\{0\}\}\) \(\cal G\) is the group of allpermutations on \(A\) which rotate the \(A_n\)'s by an angle \(\theta_n\), andsupports are finite

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