Algebra Seminar talk

Peter Holy
Compactness of strong logics

First order logic is what is used most commonly in mathematics: Given a structure with a certain domain, and with constants, relations and functions that operate on this domain, it allows us to form finitary formulas that make use of these operations, and it allows for existential and universal quantification over the domain (that is, over the elements of the domain) of our given structure. The compactness theorem for first order logic states that if T is a first order theory (in an arbitrary language with constant, relation and function symbols) such that every finite subset of T has a model, then T itself has a model. It is very interesting and often useful to also consider model theoretic properties of stronger logics. Examples of these involve logics which allow for infinitary conjunctions and disjunctions or even infinitary strings of quantification, and another prominent example is second order logic: This logic allows for quantification over subsets of a given structure. Second order logic, and most other logics that are more powerful (in terms of expressive strength) than first order logic, fail to satisfy the compactness theorem. However, classical results of Magidor closely connect compactness properties of second order logic with large cardinals, that is, principles of higher infinity that form a major backbone of modern set theoretic research: Generalizing the principle of compactness by considering subsets of a given theory T of certain infinite (rather than just finite) sizes, Magidor showed in 1971 that the least such generalized compactness cardinal for second order logic is exactly the least extendible cardinal (a prominent set theoretic principle of strong infinity), and in 1985, Makowsky showed that Vopenka's principle (another prominent set theoretic principle of strong infinity) holds if and only if every abstract logic (a very general abstract notion of logic that includes almost any logic that one might ever want to use) has such a compactness cardinal.

In this talk, I will start with a gentle introduction to the subject area, provide the relevant notions in detail, and then I will present a new type of compactness principle that we call outward compactness: The basic idea is to only consider theories T with the property that small subsets of T have models not only in our set theoretic (or, mathematical) universe, but also in suitable extensions of this universe (note that while passing to larger universes does never affect the first order properties of mathematical structures, it may, and sometimes does, affect its properties with respect to stronger logics -- for example, with respect to second order logic, because our structures may have more subsets in larger universes). We showed that these principles of outward compactness characterize a number of prominent large cardinal notions, including measurable cardinals, strong cardinals and supercompact cardinals, using compactness properties of second order logic. We also use such a principle to characterize when the class of ordinals is Woodin, using compactness properties of arbitrary abstract logics. While I will only briefly mention most of the above results, if time allows, at the end of my talk, I will try to present a proof sketch for the most basic case of measurable cardinals.

This is joint work with Philipp Lücke and Sandra Müller.