\(\mathsf{L}(\mathbb{R})\) with determinacy satisfies the Suslin hypothesis. (English) Zbl 1539.03143
Summary: The Suslin hypothesis states that there are no nonseparable complete dense linear orderings without endpoints which have the countable chain condition. \(\mathsf{ZF} + \mathsf{AD}^+ + \mathsf{V} = \mathsf{L}(\mathcal{P}(\mathbb{R}))\) proves the Suslin hypothesis. In particular, if \(L(\mathbb{R}) \vDash \mathsf{AD}\), then \(L(\mathbb{R})\) satisfies the Suslin hypothesis, which answers a question of M. Foreman [in: Logic, methodology and philosophy of science VIII. Proceedings of the eighth international congress of logic, methodology and philosophy of science, Moscow, August 17–22, 1987. Amsterdam etc.: North-Holland. 223–244 (1989; Zbl 0703.03031)].
MSC:
| 03E05 | Other combinatorial set theory |
| 03E35 | Consistency and independence results |
| 03E15 | Descriptive set theory |
| 03E45 | Inner models, including constructibility, ordinal definability, and core models |
| 03E60 | Determinacy principles |
Citations:
Zbl 0703.03031References:
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