Spring 2009

### MTHS 657 Gödel's Incompleteness Theorems

Mulvey,Irene

01/28/2009 - 05/06/2009
Wednesday 06:00 PM - 08:30 PM

Science Tower 139

In 1931, 25-year-old Kurt Gödel published the first of his remarkable theorems attacking one of the central problems in the foundations of mathematics. Typically, a branch of mathematics, (e.g., geometry) begins with some undefined terms (e.g., point, line) and some axioms (e.g., "Any two points determine a line."). The axioms, which we agree to accept as true, describe properties of the undefined terms. Within an axiomatic system, a mathematician goes about trying to prove new propositions using only the definitions, axioms, logic and previously proved propositions.

An axiomatic system is consistent provided that it is not possible to deduce two contradictory statements from the axioms. It's essential that an axiomatic system be consistent since in an inconsistent system, it is possible to deduce any proposition. An axiomatic system is complete as long as it is possible, in theory at least, to prove within the system that any given statement is either true or false. Gödel proved two things (1) any axiomatic system rich enough to include the simple laws of arithmetic of whole numbers is not complete; and (2) if an axiomatic system is consistent, then there are true statements that are not logically deducible from the axioms.

His meta-mathematical result is especially striking when placed in the context of its time. In 1900, at the Second International Congress of Mathematicians in Paris, David Hilbert proposed a list of 23 unsolved problems for the new century ("Hilbert's Problems") whose solutions would be influential in directing twentieth century mathematics. Problem 2 was to find a proof that the axioms of arithmetic are consistent. Thirty-one years later, instead of a solution, Gödel's results undermined the very preconceptions behind Hilbert's second problem.

In this course, we will begin by studying the basic notion of an axiomatic system and look at various examples. We'll cover ideas like consistency and completeness within the examples. Ultimately, we will work to understand the statement of Gödel's Incompleteness Theorems and the ideas behind their proofs.

Our main text will be the classic paperback, Gödel's Proof by E. Nagel, J. Newman, and D. Hofstadter. In addition, the instructor will have a number of supplementary texts available for short-term borrowing—one will have more technical details of the mathematics and others will be biographical works to provide context for a reclusive genius and his exceptional achievements.

Problem sets will be assigned throughout the semester. Grades will be based on problem set grades.

Irene Mulvey (B.A., Stonehill College; Ph.D., Wesleyan University) is professor of mathematics at Fairfield University. Her recent publications include "Symbolic Representation for a Class of Unimodal Cycles," Topology and Its Applications (2002), and "Multi-modal Cycles with Linear Map Having Exactly One Fixed Point," International Journal of Mathematics and Mathematical Sciences (2001). Click here for more information about Irene Mulvey.

ENROLLMENT INFORMATION

Consent of Instructor Required: No

 Format: Seminar Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
Nagel, Newman, Hofstadter, GODEL'S PROOF (NYU Press), Hardcover