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Register a new userCohen Forcing and Inner Models
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Jonas Reitz
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Given an inner model $W subset V$ and a regular cardinal $kappa$, we consider two alternatives for adding a subset to $kappa$ by forcing: the Cohen poset $Add(kappa,1)$, and the Cohen poset of the inner model $Add(kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from $V$ to fail to be as strong as that from $W$. The results are generalized to $Add(kappa,lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
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We work with symmetric inner models of forcing extensions based on strongly compact Prikry forcing to extend some known results.
These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA. Section 14 and 15 contain new results concerning the lengths of Borel hierarchies in the Cohen and random real model. Part 2 contains standard results on the theory of Analytic sets. Section 25 contains Harringtons Theorem that it is consistent to have $Pi^1_2$ sets of arbitrary cardinality. Part 3 has the usual separation theorems. Part 4 gives some applications of Gandy forcing. We reverse the usual trend and use forcing arguments instead of Baire category. In particular, Louveaus Theorem on $Pi^0_alpha$ hyp-sets has a simpler proof using forcing.
We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas with parahoric level structure. More generally our results apply to the Pappas-Zhu and Levin mixed characteristic parahoric local models, and to their equal characteristic analogues. For any such local model we prove under minimal assumptions that the entire local model is normal with reduced special fiber and, if $p>2$, it is also Cohen-Macaulay. This proves a conjecture of Pappas and Zhu, and shows that the integral models of Shimura varieties constructed by Kisin and Pappas are Cohen-Macaulay as well.
We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(lambda_i : i le alpha<aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<lambda_0)$, there is an $L_{omega_1,omega}$ -sentence $psi$ whose models form a pure AEC and (1) The models of $psi$ satisfy JEP$(<lambda_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{lambda_i^+}$ non-isomorphic maximal models of $psi$ in $lambda_i^+$, for all $i le alpha$, but no maximal models in any other cardinality; and (3) $psi$ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $aleph_0$ are at least $beth_{omega_1}$. We show that although AP$(kappa)$ for each $kappa$ implies the full amalgamation property, JEP$(kappa)$ for each kappa does not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.
Simpson showed that every countable model $mathcal{M} models mathsf{PA}$ has an expansion $(mathcal{M}, X) models mathsf{PA}^*$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a non-prime model in which the definable elements coincide with those of the underlying model. Enayat showed that this is impossible by proving that there is $mathcal{M} models mathsf{PA}$ such that for each undefinable class $X$ of $mathcal{M}$, the expansion $(mathcal{M}, X)$ is pointwise definable. We call models with this property Enayat models. In this paper, we study Enayat models and show that a model of $mathsf{PA}$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order $gamma$, if there is a model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$, then there is an Enayat model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$.
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