Fraenkel \(\cal N58\): 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\(A=\bigcup \{A_n: n\in\omega-\{0\}\}\) | Back to this models page
Description: \(\cal G\) is the group of allpermutations on \(A\) which rotate the \(A_n\)'s by an angle \(\theta_n\), andsupports are finite
When the book was first being written, only the following form classes were known to be true in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 58 |
There is an ordinal \(\alpha\) such that \(\aleph(2^{\aleph_{\alpha }})\neq\aleph_{\alpha +1}\). (\(\aleph(2^{\aleph_{\alpha}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph _{\alpha}}\).) |
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
When the book was first being written, only the following form classes were known to be false in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 126 | \(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). |
| 131 | \(MC_\omega(\aleph_0,\infty)\): 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 \(x\). |
| 154 | Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 341 | Every Lindelöf metric space is second countable. |
| 343 | A product of non-empty, compact \(T_2\) topological spaces is non-empty. |
Historical background: It was shown in Keremedis/Tachtsis [2000]that the family \(\{A_n: n\in \omega-\{0\}\}\) does not have a multiplechoice function. Thus, MC(\(\aleph_0\),\(\infty\)),Form 126, is false. Inaddition, it is shown in Keremedis/Tachtsis [2001] that 9 istrue, but 131, 154, 165, 341, and 343 are false.
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