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The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative.

Namely, there is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to the hyperreals, but even such a simple function as the sine does not extend to the surreals (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).

The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative.

Namely, there is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to hyperreal extensions of the real field, but even such a simple function as the sine does not extend to surreal extensions (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).

There is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to the hyperreals, but even such a simple function as the sine does not extend to the surreals (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).

The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine function. As far as I know the answer is negative.

Namely, there is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to the hyperreals, but even such a simple function as the sine does not extend to the surreals (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).

There is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to the hyperreals, but even such a simple function as the sine does not extend to the surreals.

There is no transfer principle in the surreals other than the one transfered from the hyperreals. Therefore it one wishes to do analysis with anything smaller than the absolutely largest class of numbers, the surreals are not an option. For example, all real functions extend to the hyperreals, but even such a simple function as the sine does not extend to the surreals (without passing via an identification of a maximal class-size surreal field and exploiting an identification of the latter with a class-size hyperreal field and importing a hyperreal transfer principle via the identification).