Strong logics explaining mathematical phenomena

Mirna Dzamonja

First order logic is the basic logic for mathematics and computer sciences. However, it plays a different role in the two subjects. In computer sciences it is often the upper limit of what is reasonable, for example, rather than studying the full FO one studies a fragment. In mathematics, the important properties of completeness and compactness of first order logic make it a wonderful tool to work with, but in practice, this tool does not have enough expressivity to explain the context in question. For example, first order logic has very little to say in the context of analysis and it is also not sufficient in order to deal with objects which are in any given sense inherently uncountable.

Our recent research has found surprising connections between these two distinct approaches to logic, the fragment one of the computer sciences and the extension one of mathematics. When one goes to certain uncountable cardinals called singular, one discovers a wealth of contexts where the computer sciences and mathematics methods overlap, and one can use computery sort of thinking to resolve problems in mathematics.