In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formulain the language of rings, there are finitely many pairs (d,μ) ∈ω×Q>0so that in any finite fieldFand for any ā ∈Fmthe size |ø(Fn,ā)| is “approximately”μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of (...) the first-order-definable sets. (shrink)

We consider low-dimensional groups and group-actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is -by-finite, and that any 2-dimensional asymptotic group is soluble-by-finite. We obtain a field-interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions.

Written in a highly personal style, Spinoza's "Ethics" presents to readers anordered vision of the universe as a unified whole--not as a lifeless world ofinnumerable separate entities. Copyright © Libri GmbH. All rights reserved.

We define an evolving Shelah‐Spencer process as one by which a random graph grows, with at each time a new node incorporated and attached to each previous node with probability, where is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah‐Spencer sparse random graphs discussed in [21] and throughout the model‐theoretic literature. The first order axiomatisation for classical Shelah‐Spencer graphs comprises a Generic Extension axiom scheme and a No Dense Subgraphs axiom (...) scheme. We show that in our context Generic Extension continues to hold. While No Dense Subgraphs fails, a weaker Few Rigid Subgraphs property holds. (shrink)