Giacomo Cozzi: Modifying the Rules of Mathematics with Extensions of Set Theory

Have you ever wanted to say things like 'Let \(\varepsilon > 0 \) be infinitely small' or 'Suppose this category is a set' but found yourself being stopped by this annoying thing called 'rules'? If Mathematics is like a game, then set theory (ZF) can be thought as a set of game rules, and in this talk we will make our own rules. We will get to know two extensions of set theory called Internal Set Theory and Feferman Set Theory. The first is an axiomatic basis of nonstandard analysis that allows for the existence infinitesimals. The latter is an attempt to solve issues of size arising when talking about categories as sets, and finally be able to talk about the category of rings without breaking into a sweat. We will also think about how these theories relate to our known mathematical universe.