Hypothesis: HR 9:
Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
Conclusion: HR 116:
Every compact \(T_2\) space is weakly Loeb. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.
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 |
Code: 3
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