2016/03/30 by Paolo Lipparini, Lipparini, Paolo, István Mező +1
Computer Science · Mathematics · #03H05 #12J15 #40A05 #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1603.09289
openalex publication_date 2016/03/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We note that if a sequence of real numbers converges to some limit, then the sequence of the corresponding strings in the surreal +,- sign expansion representation converges, for a natural notion of string convergence, to the string corresponding to the limit, modulo an infinitesimal. The corresponding statement would be obviously false if we were considering, as strings, decimal or binary representations, instead. The string limit of a possibly transfinite sequence of surreal numbers is always defined and, when considering increasing sequences of ordinals, corresponds to taking the supremum. A transfinite sum can be defined using the string limit and this sum agrees with the representation of a surreal number in Conway normal form.