There will be two mini-talks based on recent works of the speakers. Andreas Blass's invited address has been postponed.
Talk 0 (Dhruv): The Well-Foundedness of Cardinals and the Axiom of Choice
The question of whether the well-foundedness of cardinals implies the Axiom of Choice has been raised on many occasions, and remains a major open problem. In the complete absence of Choice, however, there turn out to be various notions of well-foundedness. These notions will be discussed, alongside the implications among them. Some interactions between these and other consequences of Choice will also be addressed. For example, assuming the Partition Principle, all these well-foundedness principles are provably equivalent. Furthermore, one of these principles, concerning surjections, implies the Dual Cantor-Schröder-Bernstein theorem. We end this talk by listing some open problems in this field. This talk is based on joint work with Andreas Blass.
Talk 1 (Ilir): A Gentle Walk through Function Clones
In this talk, we introduce the notion of a function clone, and then look at the lattice of all finite and countable-arity clones on 2 = {0,1}: sets of functions from 2^n to 2, n ≤ ω, which are closed under composition and include all projections. We first discuss Post's Theorem, where in 1941 he classified all finite clones over 2, then discuss our work on classifying all countable Borel clones on 2.
Talk 0 (Dhruv): The Well-Foundedness of Cardinals and the Axiom of Choice
The question of whether the well-foundedness of cardinals implies the Axiom of Choice has been raised on many occasions, and remains a major open problem. In the complete absence of Choice, however, there turn out to be various notions of well-foundedness. These notions will be discussed, alongside the implications among them. Some interactions between these and other consequences of Choice will also be addressed. For example, assuming the Partition Principle, all these well-foundedness principles are provably equivalent. Furthermore, one of these principles, concerning surjections, implies the Dual Cantor-Schröder-Bernstein theorem. We end this talk by listing some open problems in this field. This talk is based on joint work with Andreas Blass.
Talk 1 (Ilir): A Gentle Walk through Function Clones
In this talk, we introduce the notion of a function clone, and then look at the lattice of all finite and countable-arity clones on 2 = {0,1}: sets of functions from 2^n to 2, n ≤ ω, which are closed under composition and include all projections. We first discuss Post's Theorem, where in 1941 he classified all finite clones over 2, then discuss our work on classifying all countable Borel clones on 2.
| Building: | East Hall |
|---|---|
| Event Type: | Lecture / Discussion |
| Tags: | Graduate Students, Mathematics, seminar, Talk, Undergraduate Students |
| Source: | Happening @ Michigan from Student Logic and History of Math Seminar - Department of Mathematics, Department of Mathematics |
