Implications

This non-implication, Form 127 \( \not \Rightarrow \) Form 131, 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: 1040, whose string of implications is:
    9 \(\Rightarrow\) 64 \(\Rightarrow\) 127
  • 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>

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

The conclusion Form 127 \( \not \Rightarrow \) Form 131 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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