On forcing over \(L(\mathbb{R})\). (English) Zbl 07680028
Summary: Given that \(L(\mathbb{R})\models\text{ZF}+\text{AD}+\text{DC}\), we present conditions under which one can generically add new elements to \(L(\mathbb{R})\) and obtain a model of \(\text{ZF}+\text{AD}+\text{DC}\). This work is motivated by the desire to identify the smallest cardinal \(\kappa\) in \(L(\mathbb{R})\) for which one can generically add a new subset \(g\subseteq\kappa\) to \(L(\mathbb{R})\) such that \(L(\mathbb{R})(g)\models\text{ZF}+\text{AD}+\text{DC}\).
MSC:
| 03E40 | Other aspects of forcing and Boolean-valued models |
| 03E60 | Determinacy principles |
| 03E45 | Inner models, including constructibility, ordinal definability, and core models |
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