Kurt Gödel Research Center

2012 seminar talk: A spectral approach to reflection principles

Talk held by Miguel Angel Mota (KGRC) at the KGRC seminar on 2012-06-14.

Abstract

Gödel's response to the incompleteness of ZFC with respect to assertions like the Continuum Hypothesis was that one should add new natural and well justified axioms that decide these assertions. Furthermore, he suggested strong axioms of infinity, now more commonly known as large cardinal axioms, as the natural candidates to be added to ZFC. The first such new axiom of infinity is the Axiom of Inaccessibles, asserting the existence of uncountable regular, strong limit cardinals. But this axiom is only an instance of a more general class of axioms: the so called Reflection Principles. It is our aim to give a brief mathematical presentation of these principles and to suggest a philosophical justification for them. Our justification does not depend on the current idea that the universe of all sets is so complex that it can not be uniquely characterized; but rather on the intuitive fact that if a property makes a clear division between an initial segment and the global behaviour of its elements then this initial segment must be a set, since it is the obvious way to attain with the greatest certainty and obviousness ever newer number classes, and with them all the different, successive, ascending powers occurring in physical or mental nature. In this talk we will discuss (with more or less depth) the general problem of accepting new axioms, the mathematical status of set theory, the alleged paradoxes, and the philosophical standpoint of mathematical realism. We think that all these topics are related, and that they can help us to see why reflection principles must be natural axioms for set theory.