Calliope Ryan-Smith (University of Leeds) – Humanising surreal numbers

The concept of a monster model in model theory is a helpful tool for avoiding cumbersome bookkeeping and notation. Instead of needing to repeatedly take elementary extensions to realise types, or find automorphisms, one can simply say that there is a model M of a theory such that any 'small' model of that theory embeds into M, and any 'small' partial isomorphism in M extends to an automorphism. While usually 'small' means 'cardinality less than M' (and M is then taken to be 'big enough'), sometimes there are special cases in which 'small' means 'set-sized'. In particular, the surreal numbers acts as a monster model for the theory of dense linear orders, real-closed fields, and more.

I will introduce the concept of a monster model and its uses, expand somewhat on the hidden set-theoretic baggage associated with these objects, and show off the monstrosity of the surreal numbers. I will then show how, without the axiom of choice, the surreal numbers may no longer be a monster, using a simple symmetric extension argument.