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.