The successor of omega is (the equivalence class / order-type of) well-orderings that look like an infinite upward chain and then one more element above those. In general, if A is a set of ordinals that has a largest element then (A|) is the order-type of well-orderings that look like that largest element plus one more thing above it; if A is a set of ordinals that doesn’t have a largest element then (A|) is the order-type of the union of the things in A, with the understanding that if a<b with a,b in A then “corresponding” elements of a and b are identified.
(As soon as you start considering (A|B) with B nonempty, of course you are at risk of getting things that aren’t ordinals any more. But as long as you’re working with ordinals and keeping B empty, the surreal-number structure is closely related to the ordinal structure.)
It’s a while since I looked at this stuff in detail, but my recollection is that surreal numbers and nonstandard analysis don’t make very good playmates even though they are both extensions of our “usual” number systems to allow infinitesimals and infinities. One thing that makes nonstandard analysis interesting is the transfer principle (i.e., things you can say without referring to the special notions of nonstandard analysis are true about its extended number system iff they’re true about the ordinary real numbers) and so far as I know there is nothing at all like a transfer principle for the surreal numbers.
The successor of omega is (the equivalence class / order-type of) well-orderings that look like an infinite upward chain and then one more element above those. In general, if A is a set of ordinals that has a largest element then (A|) is the order-type of well-orderings that look like that largest element plus one more thing above it; if A is a set of ordinals that doesn’t have a largest element then (A|) is the order-type of the union of the things in A, with the understanding that if a<b with a,b in A then “corresponding” elements of a and b are identified.
(As soon as you start considering (A|B) with B nonempty, of course you are at risk of getting things that aren’t ordinals any more. But as long as you’re working with ordinals and keeping B empty, the surreal-number structure is closely related to the ordinal structure.)
It’s a while since I looked at this stuff in detail, but my recollection is that surreal numbers and nonstandard analysis don’t make very good playmates even though they are both extensions of our “usual” number systems to allow infinitesimals and infinities. One thing that makes nonstandard analysis interesting is the transfer principle (i.e., things you can say without referring to the special notions of nonstandard analysis are true about its extended number system iff they’re true about the ordinary real numbers) and so far as I know there is nothing at all like a transfer principle for the surreal numbers.