The Consistency of Arithmetic [pdf]

Re: making meaningful statements outside of a formal language.

Well, yes I guess that makes sense if we're talking "meaningful" in a formal sense... but then we're again stuck with the definition of what "meaningful" is.

If I don't agree with your definition of "meaningful", then why would I listen to "your" philosophy? (EDIT: This is not meant to be personally confrontational, just a blunt statement of the essential problem.)

What do you mean by "in N"?

If you mean the natural numbers then they definitely aren't "in N" since the reals already have a larger cardinality than the naturals, and the reals are a subset of the surreals.

The surreals have no relation to any of this. I'm guessing you've confused them with nonstandard natural numbers? In which case the answer is, well, what are we assuming? From the point of view of ZFC, of course there are no nonstandard naturals -- but PA can't prove this. Note that this is from the point of ZFC that PA can't prove this, since PA itself can't even formalize the notion of a nonstandard natural number (if it could, it could prove they don't exist).

In any case, surreal numbers are an entirely different system of numbers that exist in ordinary mathematics. As opposed to nonstandard naturals, which are a "what if we look at other ways of doing math?" thing.

By "first inaccessible ordinal" you mean epsilon_0? (Probably you mean it in the sense of the first one inaccessible by addition, multiplication and exponentiation.) Because there is also a "first recursively inaccessible ordinal", which is much larger (also, not recursive: it's used to construct, by collapsing, the ordinal of Delta12-CA+BI, which is also much larger though recursive).