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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.

user784623
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    You have to clearly specify a theory in question - "the theory of surreal numbers and games" isn't specific enough. We can implement the construction and analysis of surreal numbers and games, with varying levels of success, in various different axiom systems ranging from $\mathsf{RCA_0}$ (which can't prove very much) to $\mathsf{ZFC}$ (which is what's usually tacitly assumed, but is ridiculously strong proof-theoretically). – Noah Schweber May 15 '20 at 18:45
  • @NoahSchweber I have made an attempt to specify in a comment below. Re: "ridiculously strong" - do you have any good sources discusing the proof-theoretic strength of ZFC? – user784623 May 15 '20 at 20:00
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    For example, ordinal analysis has only been developed up to around $\Pi^1_2$-$\mathsf{CA_0}$ which is impossibly weak compared to $\mathsf{ZF}$; we're nowhere close to computing the proof-theoretic ordinal of $\mathsf{ZF}$. See e.g. the discussion here. – Noah Schweber May 15 '20 at 20:03

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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.

Tanner Swett
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  • If we narrow it done to numbers, would Conway's two rules (section 1.1 here) constitute as a theory in the second sense? – user784623 May 15 '20 at 19:54
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    @user784623 Technically yes, but practically no. We think of theories as being self-sufficient machines that behave in an interesting way, or which manufacture interesting products. Here's an analogy: A drill bit is technically a machine, but a drill bit all by itself doesn't do anything interesting, so it's not a very interesting machine. It's not the sort of thing we have in mind when we say the word "machine". We have to combine a drill bit with other parts (namely, a drill) in order to make an interesting machine. Likewise, those two axioms you mention are technically a theory (…) – Tanner Swett May 16 '20 at 01:40
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    (…) but those axioms all by themselves don't do anything interesting, so they're not a very interesting theory. They're not the sort of thing we have in mind when we say the word "theory". We have to combine those two axioms with other axioms (namely, a set theory) in order to make an interesting theory. – Tanner Swett May 16 '20 at 01:41