Surreal Dyadic and Real Numbers: A Formal Construction

The concept of surreal numbers, as postulated by John Conway, represents a complex and multifaceted structure that encompasses a multitude of familiar number systems, including the real numbers, as integral components. In this study, we undertake the construction of the real numbers, commencing with the integers and dyadic rationals as preliminary steps. We proceed to contrast the resulting set of real numbers derived from our construction with the axiomatically defined set of real numbers based on Conway’s axiom. Our findings reveal that both approaches culminate in the same set.