Logic

Given a topology on the natural numbers, how complicated is it to describe? To answer this question with tools from computability theory, we will restrict to the context of countable second-countable (CSC) topological spaces. One approach is to assign an index to each computable CSC space and determine the arithmetic complexity of the set of CSC spaces with some property. Another approach comes from computable structure theory; for example, given two computable copies of a CSC space, does there exist a computable homeomorphism between them? In this talk, we will explore these approaches and apply them in three running examples: the indiscrete, discrete, and initial segment topologies.

I will give a talk on several recent results regarding ultrahomogeneous graphs. A relational structure R is ultrahomogeneous if every isomorphism of finite induced substructures of R extends to an automorphism of R. We classify the finite vertex-colored oriented ultrahomogeneous graphs and the finite vertex-colored undirected graphs. The classifications comprise new general methods which govern how graphs can be combined or extended to create new ultrahomogeneous graphs. Further, we extend a classic theorem of Cameron (every 5-tuple regular graph is already ultrahomogeneous) to the setting of colored graphs.

This is joint work with Sofia Brenner, Eda Kaja, Thomas Schneider, and Pascal Schweitzer.

Fix a sequence of countable structures $\mathcal{M}_n$, for $n$ in $ℕ$, in a given first-order language $\mathcal{L}$. The reduced product of $(\mathcal{M}_n)$, denoted $\prod_n\mathcal{M}_n/\mathrm{Fin}$ is the $\mathcal{L}$-structure obtained by quotienting the product of the $\mathcal{M}_n$ by the equivalence relation of 'eventual equality'. This is similar to the ultraproduct construction, yet its theory is fairly less understood.

We consider the following question: if two reduced products $\prod_n\mathcal{M}_n/\mathrm{Fin}$ and $\prod_n\mathcal{N}_n/\mathrm{Fin}$ are isomorphic, what can be said about relations between structures we started with? Of course, one can obtain isomorphisms by taking fiberwise isomorphisms, and/or shuffling the indexes, but is that all? We discuss how, in specific interesting cases (such as fields, or certain graphs), answers to this question depend on the set theoretic ambient.

The concept of a monster model in model theory is a helpful tool for avoiding cumbersome bookkeeping and notation. Instead of needing to repeatedly take elementary extensions to realise types, or find automorphisms, one can simply say that there is a model M of a theory such that any 'small' model of that theory embeds into M, and any 'small' partial isomorphism in M extends to an automorphism. While usually 'small' means 'cardinality less than M' (and M is then taken to be 'big enough'), sometimes there are special cases in which 'small' means 'set-sized'. In particular, the surreal numbers acts as a monster model for the theory of dense linear orders, real-closed fields, and more.

I will introduce the concept of a monster model and its uses, expand somewhat on the hidden set-theoretic baggage associated with these objects, and show off the monstrosity of the surreal numbers. I will then show how, without the axiom of choice, the surreal numbers may no longer be a monster, using a simple symmetric extension argument.

Fundamental to the practice of logic is the dogma regarding the first order/second order logic distinction, namely that it is ironclad. Was it always so? The emergence of the set theoretic paradigm is an interesting test case. Early workers in foundations generally used higher order systems in the form of type theory; but then higher order systems were gradually abandoned in favor of first order set theory—a transition that was completed, more or less, by the 1930s.

As for logic in general, the concept of a logic being first order is not only about whether the variables range over the elements of a given domain, or over sets of elements, or over sets of sets of elements, and so on; it is also, I suggest, about the context.

Of course, set theory is a theory and second order logic is a logic, at least that is the common understanding. However if one cares to view set theory as a logic—and if we do think of set theory as a logic, it is a logic with the cumulative hierarchy 𝑉 as its standard (class) model—then set theory turns out to be a stronger logic than second order logic. This is perhaps as it should be, given that the latter restricts the domain of quantifiable objects to those generated by (at most) a single iteration of the power set operation, while set theory allows for arbitrary iterations of the power set operation.

This talk is based on the forthcoming paper "How first order is first order logic?" by J. Kennedy and Jouko Väänänen for The Oxford Handbook of Philosophy of Logic. Editors: Elke Brendel, Massimiliano Carrara, Filippo Ferrari, Ole Hjortland, Gil Sagi, Gila Sher, Florian Steinberger, Oxford University Press.