Cool Logic

Paula Henk (ILLC)

What We Know but Peano Arithmetic Doesn't - Modal Logics of Nonstandard Notions of Provability

November 13th at 17:30, in ILLC Seminar Room (F1.15)

This talk will take you to an excursion inside Peano Arithmetic (PA), with a particular emphasis on how what we encounter there differs from our ordinary world outside PA.

A well-known example is provided by Gödel's Second Incompleteness Theorem: while we know that PA is consistent, once we step into the world inside PA we cannot be sure of this anymore.

We shall see more interesting examples of facts that are known to us, but not to PA. I will introduce three nonstandard notions of provability. Parikh provability involves an additional rule, which sometimes allows us to obtain much shorter proofs. Slow provability on the other hand makes it harder to obtain proofs. Feferman provability is set up in such a way that it sometimes leads people to think that they have disproved Göodel's Theorem (I will explain why they are wrong).

For us, it will be rather easy to see that each of these notions of provability actually coincides with ordinary provability (in PA). However, once we step into the world inside PA, they all seem rather different from each other! But how exactly do then all these notions look like through the eyes of PA? As it turns out, this question can be given a very neat answer by using modal logic. I will outline the basic ideas behind this undertaking, pursued in the field of provability logic.

The talk does not presuppose specialized knowledge about Peano Arithmetic, and is meant to be accessible to everyone at the ILLC.