Surreal numbers

The notion of infinity has always been in the center of attention in math and philosophy. Mathematicians have worked out a number of competing theories to do calculations with infinitely big (and infinitely small) quantities. One of these approaches is the theory of surreal numbers, where we can add, subtract, multiply and divide different infinities. One can have a lot of fun in this theory.

I find very interesting the strange world of the surreal numbers, so I started to do some research in this area. It is hard to publish papers on this very special topic in standard journals, and my results are not deep, so I decided to put my results here on my webpage instead. (For the very same reason, my presentation is sloppy sometimes. Sorry for that. ;))

Limits of surreal sequences

For a long time it had not been clear how to define limits of surreal number sequences. In a short manuscript I explain when should we say that a surreal number sequence is convergent. This work was subsequently developed further through a collaboration with Prof. Paolo Lipparini.

The idea of this convergence notion came after I had given a talk at Researcher's Night in Hungary in 2011.

The back-step operator

To facilitate the construction of "very small big numbers", I defined the back-step operator. With this operator, we can more easily imagine and describe strange surreals like omega^(epsilon^omega).

Omega half

It makes no sense to talk about the cardinal aleph_1/2 in classical set theory, because aleph_0 and aleph_1 are two consecutive cardinals. But it does make sense to talk about surreal numbers which are greater than any countable ordinal but smaller than any uncountable one. They are, in some sense, half in between aleph_0 and aleph_1. In this short note we define the "simplest" among such surreals, omega_1/2.