> Whereof one cannot speak, thereof one must be silent.
did not get a citation. Perhaps its so well known among its target audience that none was considered necessary. Since HN appeals to a broader audience, I'll supply the missing citation: It's from Wittgenstein's Tractatus - the very last line in fact. The usage is apt - as far as I can tell, Wittgenstein was making the same point about the difficult - even impossibility - of 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.)
I'd love to read a similar explanation about surreal numbers. Are surreal numbers in N or not? How do we know? (In my limited understanding, they're not easily excluded.)
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.
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.