Computability and Differential Fields: a Tutorial

Computability theory applies a rigorous definition of the notion of an algorithm to determine which mathematical functions can be computed and which cannot. The main concepts in this area date back to Alan Turing, who during the 1930’s gave the definition of what is now called a Turing machine, along with its generalization, the oracle Turing machine. In the ensuing seventy years, mathematicians have developed a substantial body of knowledge about computability and the complexity of subsets of the natural numbers. It should be noted that for most of its history, this subject has been known as recursion theory ; the terms computable function and recursive function are to be treated as interchangeable. Computable model theory applies the notions of computability theory to arbitrary mathematical structures. Pure computability normally considers functions from N to N, or equivalently, subsets of finite Cartesian products N × · · · × N. Model theory is the branch of logic in which we consider a structure (i.e. a domain of elements, with appropriate functions and relations on that domain) and examine how exactly the structure can be described in our language, using symbols for those functions and relations, along with the usual logical symbols such as negation, conjunction, (∃x), and (∀x). To fit this into the context of computability, we usually assume that the domain