Department of Mathematics
More than the sum of its parts
Bernhard Koenig
Boise State University
We consider a couple of interesting phenomena in combinatorial set theory concerning trees (a tree is a partial ordering with the property that the predecessors of every point form a linear well- ordering). We ask the following question: assume we are given two trees S and T such that all proper initial segments of S are isomorphic to proper initial segments of T and vice versa (we call S and T "locally isomorphic" in this case). Does this mean that S and T are isomorphic?
It seems paradoxical to have two trees S and T that are locally isomorphic but not isomorphic, since this would mean that they are constructed using the very same building blocks, yet they would be different. We present a couple of results (some are classical, some are more recent results of the speaker) that show that the question can have different answers, depending on the height of the trees and on the axioms of set theory.
All interested persons are welcome.