Joel David Hamkins - Apr 27, 2015 - Morning Session (Part 1 of 2)

Afternoon Session (Part 2 of 2):    • Joel David Hamkins - Apr 27, 2015 - Aftern...   Title: The continuum hypothesis and other set-theoretic ideas for non-set-theorists Abstract: The background talk will discuss and prove the theorem of Cantor-Bendixson that every closed set of reals is the union of a countable set and a closed set with no isolated points. The lecture will explain how this result led to Cantor's development of the ordinal numbers and how it establishes that closed sets of reals are either countable or have the same size as the set of all real numbers. We'll see that there are closed sets of arbitrary countable depth in the sense of their analysis. Then ordinals will be discussed: the first uncountable ordinal, the long line, and time permitting Suslin's hypothesis concerning the possibility to characterize the real numbers in terms of its order structure. This will bring the discussion to the topic of the afternoon's session, Cantor's question: Is there an uncountable subset of real numbers whose size is strictly smaller than the size of all real numbers? The "Continuum Hypothesis." is the truth of the negative response to this question. The organizers thought the discussion just before or during lunch might also touch on the role of set theory and logic s in the foundations of mathematics. Granted that set theory leads to a rich and beautiful part of mathematics, one wonders if there might be something to be gained by seriously developing the logic of a categorical foundation for mathematics which underlies the theory of sets in its role as providing a basis for mathematics. The speaker is prepared to discuss the Feferman theory in comparison to Grothendieck's axiom of universes, and other issues relating set theory to category theory. CUNY Einstein Mathematics Seminar: http://goo.gl/MsQrHq