Past Logic Seminar

23 October 2014
17:30
Volker Halbach
Abstract

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

16 October 2014
17:30
Mario Edmundo
Abstract

The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimals sheaves, which might be useful on it own. 

19 June 2014
17:15
Jamshid Derakhshan
Abstract
This is joint work with Angus Macintyre. We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p. To prove this we prove universal definability of the valuation rings of such fields using work of Cluckers-Derakhshan-Leenknegt-Macintyre on existential definability, quantifier elimination of Basarab-Kuhlmann for valued fields in a many-sorted language involving higher residue rings and groups, a model completeness theorem for certain pre-ordered abelian groups which generalize Presburger arithmetic (we call finite-by-Presburger groups), and an interpretation of higher residue rings of such fields in the higher residue groups.
5 June 2014
17:15
Charlotte Kestner
Abstract
I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. This is joint work with William Anscombe.
29 May 2014
17:15
Andrew Brooke-Taylor
Abstract
<p>Cardinal characteristics of the continuum are (definitions for) cardinals that are provably uncountable and at most the cardinality c of the reals, but which (if the continuum hypothesis fails) may be strictly less than c.&nbsp; Cichon's diagram is a standard diagram laying out all of the ZFC-provable inequalities between the most familiar cardinal characteristics of the continuum.&nbsp; There is a natural analogy that can be drawn between these cardinal characteristics and highness properties of Turing oracles in computability theory, with implications taking the place of inequalities.&nbsp; The diagram in this context is mostly the same with a few extra equivalences: many of the implications were trivial or already known, but there remained gaps, which in joint work with Brendle, Ng and Nies we have filled in.</p>
22 May 2014
17:15
Will Anscombe
Abstract
<p>A 1-dimensional asymptotic class (Macpherson-Steinhorn) is a class of finite structures which satisfies the theorem of Chatzidakis-van den Dries-Macintyre about finite fields: definable sets are assigned a measure and dimension which gives the cardinality of the set asymptotically, and there are only finitely many dimensions and measures in any definable family. There are many examples of these classes, and they all have reasonably tame theories. Non-principal ultraproducts of these classes are supersimple of finite rank.<br /><br /> Recently this definition has been generalised to `Multidimensional Asymptotic Class' (joint work with Macpherson-Steinhorn-Wood). This is a much more flexible framework, suitable for multi-sorted structures. Examples are not necessarily simple. I will give conditions which imply simplicity/supersimplicity of non-principal ultraproducts.<br /><br /> An interesting example is the family of vector spaces over finite fields with a non-degenerate bilinear form (either alternating or symmetric). If there's time, I will explain some joint work with Kestner in which we look in detail at this class.</p>
13 March 2014
17:15
Colin McLarty
Abstract
Several number theorists have stressed that the proofs of FLT focus on small concrete arithmetically defined groups rings and modules, so the steps can be checked by direct calculation in any given case. The talk looks at this in relation both to Hilbert's idea of contentual (inhaltlich) mathematics, and to formal provability in Peano arithmetic and other stronger and weaker axioms.
27 February 2014
17:15
Kentaro Fujimoto
Abstract
Formal truth theory sits between mathematical logic and philosophy. In this talk, I will try to give a partial overview of formal truth theory, from my particular perspective and research, in connection to some areas of mathematical logic.
13 February 2014
17:15
Philip Welch
Abstract
It is well known that infinite perfect information two person games at low levels in the arithmetic hierarchy of sets have winning strategies for one of the players, and moreover this fact can be proven in analysis alone. This has led people to consider reverse mathematical analyses of precisely which subsystems of second order arithmetic are needed. We go over the history of these results. Recently Montalban and Shore gave a precise delineation of the amount of determinacy provable in analysis. Their arguments use concretely given levels of the Gödel constructible hierarchy. It should be possible to lift those arguments to the amount of determinacy, properly including analytic determinacy, provable in stronger theories than the standard ZFC set theory. We summarise some recent joint work with Chris Le Sueur.

Pages