Implications

This non-implication, Form 404 \( \not \Rightarrow \) Form 328, 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: 10048, whose string of implications is:
    9 \(\Rightarrow\) 404
  • A proven non-implication whose code is 3. In this case, it's Code 3: 171, Form 9 \( \not \Rightarrow \) Form 131 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 131 <p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7560, whose string of implications is:
    328 \(\Rightarrow\) 126 \(\Rightarrow\) 131

The conclusion Form 404 \( \not \Rightarrow \) Form 328 then follows.

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

Name Statement
\(\cal N55\) Keremedis/Tachtsis Model: The set of atoms \(A=\bigcup \{A_n: n\in \omega\}\), where \(A_n=\{a_{n,x}: x\in B(0,\frac1n)\}\) and \(B(0,\frac1n)= \{x: \rho(x,0)=\frac1n\}\), where \(\rho\) is the Euclidean metric The group of permutations \(\cal G\), is the group of all rotations of the \(A_n\) through an angle \(\theta\in [0,2\pi)\), and supports are finite
\(\cal N58\) 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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