Fraenkel \(\cal N57\): The set of atoms \(A=\cup\{A_{n}:n\in\aleph_{1}\}\), where\(A_{n}=\{a_{nx}:x\in B(0,1)\}\) and \(B(0,1)\) is the set of points on theunit circle centered at 0 | Back to this models page
Description: The group of permutations \(\cal{G}\) is thegroup of all permutations on \(A\) which rotate the \(A_{n}\)'s by an angle\(\theta_{n}\in\Bbb{R}\) and supports are countable
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 |
|---|---|
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 340 | Every Lindelöf metric space is separable. |
| 341 | Every Lindelöf metric space is second countable. |
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 |
|---|---|
| 173 | \(MPL\): Metric spaces are para-Lindelöf. |
| 232 | Every metric space \((X,d)\) has a \(\sigma\)-point finite base. |
| 381 | DUM: The disjoint union of metrizable spaces is metrizable. |
| 382 | DUMN: The disjoint union of metrizable spaces is normal. |
Historical background: It is shown in Dela Cruz/Hall/Howard/Keremedis\slash Rubin [2002a] that in this model8, 43, 340, and 341 are true, while 173, 232, 381, and 382 are false.
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