Mathematics Graduate Student Colloquium

MGSC Website: http://math.uci.edu/~mgsc/

Goodstein's theorem, Euclid's fifth postulate, and independence proofs

Kyriakos Kypriotakis
Wednesday, October 25, 2006
4:00 pm - 4:50 pm
PSCB 120

Talk Abstract:

In mathematical logic, a sentence is called independent of a given set of axioms T if T neither proves nor refutes that sentence. Euclid's fifth postulate and Goodstein's theorem are two such sentences. Goodstein's theorem is a statement about the natural numbers that is undecidable in Peano arithmetic (PA), but can be proven to be true using the axiom system of set theory (ZFC). Euclid's fifth postulate says that two parallel lines do not intersect. We will prove the independence of the second (there is a surprisingly easy proof) and demonstrate the power of the Goodstein's theorem (i.e., explain how it gives us the relative consistency of PA; a set of axioms T is called consistent if T does not lead to any contradictions.)

About the Speaker:

Kyriakos Kypriotakis is a fifth-year graduate student here at UCI. He earned his B.S. in mathematics at the U. of Salonica and his M.S. in mathematical logic at the U. of Athens in 2002.

Refreshments:

Pizza will be served after the talk.