**UM Math Graduate Students Seminar**

Alex Lazar

* University of Miami *

**will present**

**Set theory and Aronszajn Trees**

Friday, February 6, 2015, 3:30pm

Ungar Building Room 402

**Abstract: **

In graph theory,König's Lemma states that any tree with countably many vertices in which every vertex has finite degree must contain an infinite path. The proof of this result is simple, and leads to the natural question of whether analogous results hold for trees with larger sets of vertices. In the 1930s, Aronszajn proved that the analogous result does *not* hold for trees whose vertex set has the cardinality of the continuum by constructing an object that we now call an Aronszajn Tree. In this talk, we will discuss some basic set-theoretic notions such as cardinal and ordinal numbers, and use them to construct an Aronszajn Tree by transfinite induction

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