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Graduate Seminar

Michael Klipper
Carnegie Melllon University
Title: Compactness and generating counterexamples

Abstract: There are many innocuous mathematical statements which, upon further inspection, have surprising depth. For example, consider the statement {Every natural number is the sum of finitely many copies of 1.} Many mathematicians do not give this a second thought, viewing this as the definition of N. However, when we make this definition more formal, say via induction, we find that there is a lot of logical complexity:

{N is the set such that (1) 0 is in N, (2) for all n in N, n+1 is in N, and (3) for all sets X satisfying (1) and (2), N is contained in X.}

In particular, this definition quantifies over individual numbers (in part (2)) and also quantifies over SETS (in part (3)). Logicians would say this definition is a third-order statement. More often than not, we aim to have definitions which only quantify over individual objects, i.e. first-order definitions. Such definitions are easier to reason about. This leads us to ask questions like: if we use only first-order statements, what can we still guarantee is true about N? In this talk, I will introduce some notions of model theory to illustrate that actually, surprisingly few {obvious} properties ought to hold when you restrict yourself to only first-order statements!

Date: Tuesday, September 28, 2010
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Daniel Spector