What is the proof-theoretic strength of the theory of Surreal numbers & games?

Question

I was reading about ordinal analysis & this made me wonder what is the proof-theoretic strength of the theory of Surreal numbers & games?

Hopefully this isn't a nonsensical question.

Answer 1

The word "theory" in mathematics has two different meanings; it sounds like you may be confusing the two.

Note that the two meanings are mostly unrelated; there aren't really any similarities between the two meanings.

In one sense, the word "theory" means a particular area of study within mathematics. For example, the study of whole numbers and how they relate to each other is called "number theory".

In the other sense, the word "theory" means a collection of well-formed formulas that are considered to be axioms. In other words, it's a particular, exact list of mathematical sentences, written entirely in symbols. Here's an example of a theory:

$$\forall x \forall y \forall z \quad [[[\mathrm{S}x]y]z] = [[xz][yz]] \\ \forall x \forall y \quad [[\mathrm{K}x]y] = x$$

(Most theories that mathematicians study are much more complicated than this.)

Since theories, in this sense, are exactly defined lists of symbols, they are also mathematical objects themselves, and a theory has various mathematical properties that we can study.

In particular, a theory, in the second sense, has a particular mathematical property called its proof-theoretic strength.

The theory of surreal numbers and games is a theory in the first sense: it's a particular area of study within mathematics. I'm not aware of any theory in the second sense which could reasonably be called "the theory of surreal numbers and games". If you want to study surreal numbers and games, a good theory (in the second sense) to use is VBG set theory.