The 2017 Lindström Lectures was delivered by Albert Visser, Emeritus Professor of Philosophy at Utrecht University, where he served as professor of logic, philosophy of mathematics, and epistemology in the period 1998–2016. He obtained his Ph.D. in 1981 under the direction of Dirk van Dalen at Utrecht University, the Netherlands. His research centers on arithmetical theories, interpretability, constructivism, foundations of mathematics and topics in the philosophy of language. He is a member of the Royal Netherlands Academy of Arts and Sciences, and the editorial board of the Notre Dame Journal of Formal Logic.
Per Lindström’s work on interpretations has great beauty. He was a grand master of dazzling diagonal arguments. In this talk we will explain the basic setting underlying Per’s work. We introduce the notion of interpretation and provide some examples of interpretations. We show how, in the context of arithmetic, the notion of interpretability has an almost unrecognizable equivalent. This equivalence is known as the Orey–Hájek Characterization.
We will discuss some results of Per and have a look at further developments.
We study the Second Incompleteness Theorem, G2, in the Feferman-style. This means that we work with a fixed provability-predicate but allow the representations of the axiom set to vary. Feferman observed that the axiom set of Peano Arithmetic, PA, has a Π01-representation for which PA proves its own consistency.
We isolate a condition that Feferman’s example fails to satisfy. This condition gives a reasonably general version of G2. We show that this version yields a proof of G2 for Σ01-semi-numerations of the axiom set which works even if the theory itself is not recursively enumerable. We discuss an interesting example that illustrates that we may have G2 even in the absence of the Löb conditions.