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Graduate Seminar

Will Gunther
Carnegie Mellon University
Title: Arithmetic and incompleteness

Abstract: In the early 20th century, Hilbert's Program was attempting to re-formalize mathematics in a "simple" system of axioms capable of expressing and proving all mathematics. In 1931, Austrian logician Kurt Godel demonstrated this program would be unsuccessful by proving that any consistent system which is "sufficiently powerful" and described by a simple set of axioms is not capable of proving all true arithmetic statements. This is Godel's first incompleteness theorem.

In this talk, a sketch of the proof will be given. Focus will be on the power of primitive recursive functions to code information. To this end, we will prove Godel's Beta Function Lemma. With this tool in hand, we will show that arithmetic is "self-aware", which will give us an example of a true statement with no proof in any nice system of arithmetic (like Peano Arithmetic).

Date: Thursday, September 26, 2013
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Brian Kell