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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 thirdorder statement. More often than not, we aim to have definitions which only quantify over individual objects, i.e. firstorder definitions. Such definitions are easier to reason about. This leads us to ask questions like: if we use only firstorder 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 firstorder statements! Date: Tuesday, September 28, 2010 Time: 5:30 pm Location: Wean Hall 8220 Submitted by: Daniel Spector 