Logic Workshop

We present two different elementary algebraic constructions that are as complicated as possible and whose complexity vastly exceeds those typically found in the elementary algebra literature. The first is the prime radical of a noncommutative ring, while the second is the radical of a module. These constructions contrast similar constructions in more familiar contexts that we will also mention along the way. We will spend most of our time describing how to construct radicals that are as complicated as possible from a computability point of view.

A Turing degree is low for isomorphism if whenever it can compute an isomorphism between two countably presented structures, there is already a computable isomorphism between them and thus there is no need to use the degree as an oracle at all. I will discuss the types of degrees that are low for isomorphism and the extent to which this class of degrees has the same properties as other lowness classes.

This work is joint with Reed Solomon.

In this talk we revisit the problem of describing the 'finer' structure of geometrically trivial strongly minimal sets in $DCF_0$. In particular, I will explain how recent work joint with Guy Casale and James Freitag on automorphic functions, has lead to intriguing questions around the $\omega$-categoricity conjecture. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular $j$-function. I will explain how their counter-example fits into the larger context of arithmetic automorphic functions and has allowed us to 'propose' refinements to the original conjecture.

We prove that $\omega_1$ cannot be embedded into any preordering $\Sigma$-definable with parameters in the hereditarily finite superstructure over the ordered field of real numbers, HF(R). As corollaries, we obtain characterizations of $\Sigma$-presentable ordinals and Gödel constructive sets of kind $L_\alpha$. It also follows that there are no $\Sigma$-presentations for structures of $T$-, $m$-, $1$-, and $tt$-degrees over HF(R).

Our talk concerns axiomatic theories of truth predicates. They are theories obtained by adding to Peano Arithmetic (${\rm PA}$) a fresh predicate $T(x)$ with the intended reading '$x$ is (a code of) a true sentence in the language of arithmetic' together with some axioms governing newly added predicate.

The canonical example of such a theory is ${\rm CT}^-$ (Compositional Truth). Its axioms state that the truth predicate is compositional. For instance, a conjunction is true iff both conjuncts are. If we add to ${\rm CT}^-$ full induction in the extended language, we call the resulting theory ${\rm CT}$.

It is easy to check that ${\rm CT}$ is not conservative over ${\rm PA}$, i.e., it proves new arithmetical sentences. On the other hand, by a nontrivial theorem of Kotlarski, Krajewski, and Lachlan, ${\rm CT}^-$ extends ${\rm PA}$ conservatively.

In our talk, we will discuss results on the strength of theories between ${\rm CT}^-$ and ${\rm CT}$. It turns out that the natural axioms concerning purely truth theoretic properties of the newly added predicate (as opposed to axiom schemes which are consequences of induction in more general context) are typically either conservative or exactly equal to ${\rm CT}_0$, the theory of compositional truth with $\Delta_0$-induction. Thus ${\rm CT}_0$ turns out to be a surprisingly robust theory and, arguably, the minimal 'natural' non-conservative theory of truth.

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). We extend it to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces, which implies, e.g., several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.