Logic Seminar-Second-order logic: syntax, semantics, and a scintilla of predicativity

Time: Thursday, December 1, 11:00 am-12:00 noon

Place: Smith 120

Abstract: We will discuss second-order logic, a powerful system of mathematical logic that is often understudied today.  Contemporary mathematics is founded on axiom systems formulated in first-order logic, which allows sentences to quantify only over objects in the domain of discourse.  But historically, the foundations of mathematics were originally studied by Frege, Russell, and others using second-order logic, which allows sentences to quantify not only over objects but also over sets, relations, and functions.  We will explore the syntactic and semantic properties of second-order logic, and compare its expressive and deductive power with first-order logic.  In the final part of the talk, we will describe a modification of second-order logic which conforms to predicativism, a philosophy of mathematics which is fastidious about avoiding circular definitions.