Logic Seminar-Power and Limitations of Second-Order Logic

Time: Thursday, December 8, 11:00 am--12:00 noon

Place: Smith 120

Speaker:  Keshav Srinivasan, GWU

Title: Power and Limitations of Second-Order Logic

Abstract: We will discuss the expressive and deductive power of second-order logic, a system of mathematical logic often understudied today. We will review the limitative theorems that apply to first-order logic, which only allows quantification over elements, and ascertain the extent to which they do or do not apply to second-order logic, which allows quantification over sets, functions, and relations. We will examine a number of second-order axiom systems, including second-order Peano Arithmetic, the theory of Dedekind-complete ordered fields, and Morse-Kelley set theory, and explain their implications for the nature of second-order logic. In the final part of the talk, we will describe how Bertrand Russell and others modified second-order logic to avoid impredicativity, a form of circular definition that is pervasive in contemporary mathematics.