Recursive surreal numbers (Q750440)

The recursive reals and the constructive ordinals have been extensively studied in the past. Since the surreal numbers contain both the reals and the ordinals it is natural to unite the two subjects by means of the notion of a recursive surreal number. The author does this in two ways: first sign sequences and secondly using cuts. For the case of sign sequences the ordinals obtained are precisely the constructive ordinals and the reals obtained are precisely the recursive reals. Unfortunately addition and multiplication cannot be effectively computed. In the case of cuts, the operations can be effectively computed. Finally the author shows that the set of recursive surreals obtained using cuts properly contains the corresponding set using sign sequences. He does this by constructing a surreal of length \(\omega\cdot 2\) which is in the former but not in the latter set and uses this surreal to show that the former set contains a real number which is not recursive.