Sacks Forcing and the Shrink Wrapping Property

We consider a property stronger than the Sacks property, called the shrink wrapping property, which holds between the ground model and each Sacks forcing extension. Unlike the Sacks property, the shrink wrapping property does not hold between the ground model and a Silver forcing extension. We also show an application of the shrink wrapping property.Comment: 16 page

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Definable maximal discrete sets in forcing extensions

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $\Delta^1_2$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in [5] by showing that in the Sacks and Miller extensions there is a $\Pi^1_1$ maximal orthogonal family ("mof") of Borel probability measures on Cantor space. A similar result is also obtained for $\Pi^1_1$ mad families. By contrast, we show that if there is a Mathias real over $L$ then there are no $\Sigma^1_2$ mofs.Comment: 16 page

The complexity of classification problems for models of arithmetic

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.Comment: 15 page

Kinematic Diffraction from a Mathematical Viewpoint

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.Comment: 30 pages, invited revie

Saccharinity

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

We consider the following dichotomy for ∑02 finitary relations R on analytic subsets of the generalized Baire space for k: either all R-independent sets are of size at most k, or there is a k-perfect R-independent set. This dichotomy is the uncountable version of a result found in [W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), 619-623] and in [S. Shelah, Fund. Math. 159 (1999), 1-50]. We prove that the above statement holds if we assume ◊k and the set-theoretical hypothesis I-(k), which is the modification of the hypothesis I(k) suitable for limit cardinals. When K is inaccessible, or when R is a closed binary relation, the assumption ◊k is not needed. We obtain as a corollary the uncountable version of a result by G. Sági and the first author [Logic J. IGPL 20 (2012), 1064-1082] about the k-sized models of a ∑11(Lk+k)-sentence when considered up to isomorphism, or elementary embeddability, by elements of a Kk subset of kK. The elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving Lλμ for ω≤μ≤λ≤κ and finite variable fragments of these logics