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It is a theorem of Woodin that if there is a proper class of Woodin cardinals, then the theory of $L(\mathbb{R})$ can not be changed by forcing. Since forcing and large cardinals are essentially our only means for establishing independence results, this can be interpreted as saying that the theory of $L(\mathbb{R})$ is immune to independence phenomena (except for that which G\" odel's theorem imposes). Here $L(\mathbb{R})$ is the smallest model of ZF which contains all of the reals. $L(\mathbb{R})$ does satisfy the Axiom of Dependent Choice under this assumption, as well as the Axiom of Determinacy. Most of not all theorems in real and complex analysis, measure theory (in the setting of standard Borel space), manifolds, geometry, and number theory can be regarded as statements about what is true in $L(\mathbb{R})$. It is the ideal model in which to study descriptive set theory. Uncountable sets and cardinals, however, often behave strangely in this model (largely because of the influence of the Axiom of Determinacy). For instance, in $L(\mathbb{R})$ and under the above assumptions, $\omega_1$ and $\omega_2$ are measurable cardinals, $\omega_n$ is a singular cardinal for each $n > 2$, there is no uncountable well orderable set of reals, and there are no non-principle non-principal ultrafilters on $\omega$.
Ironically, Woodin's theorem was the culmination of several deep results concerning iterated forcing, the combinatorics of $\omega_1$, the study of large cardinals, and the fine structure of inner models generalizing $L$.
It is a theorem of Woodin that if there is a proper class of Woodin cardinals, then the theory of $L(\mathbb{R})$ can not be changed by forcing. Since forcing and large cardinals are essentially our only means for establishing independence results, this can be interpreted as saying that the theory of $L(\Rbb)$ L(\mathbb{R})$is immune to independence phenomena (except for that which G\" odel's theorem imposes). Here$L(\mathbb{R})$is the smallest model of ZF which contains all of the reals.$L(\mathbb{R})$does satisfy the Axiom of Dependent Choice under this assumption, as well as the Axiom of Determinacy. Most of not all theorems in real and complex analysis, measure theory (in the setting of standard Borel space), manifolds, geometry, and number theory can be regarded as statements about what is true in$L(\mathbb{R})$. It is the ideal model in which to study descriptive set theory. Uncountable sets and cardinals, however, often behave strangely in this model (largely because of the influence of the Axiom of Determinacy). For instance, in$L(\mathbb{R})$and under the above assumptions,$\omega_1$and$\omega_2$are measurable cardinals,$\omega_n$is a singular cardinal for each$n > 2$, there is no uncountable well orderable set of reals, and there are no non-principle ultrafilters on$\omega$. Ironically, Woodin's theorem was the culmination of several deep results concerning iterated forcing, the combinatorics of$\omega_1$, the study of large cardinals, and the fine structure of inner models generalizing$L$. 2 fixed grammar in second paragraph It is a theorem of Woodin that if there is a proper class of Woodin cardinals, then the theory of$L(\mathbb{R})$can not be changed by forcing. Since forcing and large cardinals are essentially our only means for establishing independence results, this can be interpreted as saying that the theory of$L(\Rbb)$is immune to independence phenomena (except for that which G\" odel's theorem imposes). Here$L(\mathbb{R})$is the smallest model of ZF which contains all of the reals.$L(\mathbb{R})$does satisfy the Axiom of Dependent Choice under this assumption, as well as the Axiom of Determinacy. Most of not all theorems in real and complex analysis, measure theory (in the setting of standard Borel space), manifolds, geometry, and number theory can be regarded as statements about what is true in$L(\mathbb{R})$. It is the ideal model in which to study descriptive set theory. Uncountable sets and cardinals, however, often behave strangely in this model (largely because of the influence of the Axiom of Determinacy). For instance, in$L(\mathbb{R})$and under the above assumptions,$\omega_1$and$\omega_2$are measurable cardinals,$\omega_n$is a singular cardinal for each$n > 2$, there is no uncountable well orderable set of reals, and there are no non-principle ultrafilters on$\omega$. It should be noted, however Ironically, that ironically Woodin's theorem was the culmination of several deep results concerning iterated forcing, the combinatorics of$\omega_1$, the study of large cardinals, and the fine structure of inner models generalizing$L\$.