The CEMS.UL - Center for Mathematical Studies is promoting the seminar "Reflexivity of Omega-Consistency in a General Setting", with the participation of Paulo Guilherme Santos (IPMA Lisboa and CEMS.UL)
Abstract: How much can a mathematical theory certify about its own reliability? This question sits at the heart of metamathematics, and it grows especially delicate once we ask a theory to vouch for the consistency of its own fragments. A theory is called reflexive when it can prove the consistency of each of its finitely axiomatized sub-theories, and essentially reflexive when this ability survives the addition of any finite batch of new axioms. Peano Arithmetic, together with the standard set theories, enjoys this remarkable property.
In this talk we move beyond ordinary consistency to a richer family of reliability statements modelled on omega-consistency — the property that a theory never proves a claim for every individual natural number while simultaneously proving that some number fails it. We show that essentially reflexive theories are reflexive not only for ordinary consistency but also for its restriction to formulas of bounded quantifier complexity. Partial truth predicates take centre stage along the way, and they yield a strikingly short proof that a natural theory of truth built over Peano Arithmetic already establishes the full omega-consistency of arithmetic.
We then introduce a new notion, which we call induction-consistency: the property that a theory does not refute full induction. Although it looks unrelated to omega-consistency at first glance, it turns out to be an instance of the very same phenomenon. To make this precise, we develop a single general framework — a parametrized form of consistency assembled from an array of quantifiers and a pair of auxiliary terms — that captures omega-consistency, its bounded and uniform variants, and induction-consistency all at once. A common thread of reasoning, refined throughout the talk, culminates in one master theorem from which the earlier results all flow as special cases. We also pin down exactly which reflection principles, and which concrete fragments of arithmetic, are needed to secure each form of consistency.