Foundations for the Analysis of Surreal-Valued Genetic Functions

In this thesis we systematize earlier results from the literature of functions on surreal num- bers and consider the generalization of results of Ehrlich and van den Dries regarding models of real-closed fields with exponentiation to the wider class of genetic functions, which includes many examples of interest such as the class of restricted analytic functions, exp, and log, as well as the ω map and other recursively definable functions. We do so by first amending the construction of arbitrary genetic functions found in the literature, so that we may properly compose functions, and so that one can easily recover the definition of exp. We then analyze our newly proposed inductive construction with two natural notions of complexity - that of generation, which tracks the dependence on earlier genetic functions, and that of Veblen rank, which describes the complexity of subtrees closed under a genetic function - in order to characterize the ordinals α such that the surreal numbers below height α will correspond to models satisfying the cofinality conditions and the axioms of real closed fields. After recovering fundamental analytic results for general surreal-valued functions, we further prove that every genetic function has a Veblen rank corresponding to an ordinal, and that our notion of Veblen rank behaves well under addition, multiplication, and composition, and in turn can be extended to arbitrary sets closed under said operations. In particular, the Veblen rank of a genetic function g identifies the largest subclass of epsilon numbers α such that sets of surreal numbers of height below α form a real closed field closed under g. From this, we establish many important functions, such as exp and log will have minimal Veblen rank, while the lambda and kappa maps used to define the Berarducci-Mantova derivative have non-trivial Veblen rank. As a further consequence of our Veblen rank bound, we establish that every entire genetic function is strictly tame in the sense of Fornasiero [4]. Afterwards, with G denoting a set of genetic functions, we proceed to define a general first order theory whose models are G-closed fields satisfying fundamental first order properties used to define each genetic function.