Models of Reverse Mathematica
University of Pennsylvania, USA
Reverse Mathematics is concerned with measuring the strength of mathematical theorems in the context of second order arithmetic---a theory which discusses both numbers and sets of numbers. In most cases we can separate the "first-order" consequences of a theorem---how closely the numbers in the theory must resemble the actual natural numbers---from the "second-order" consequences---which sets must exist for the theorem to be true. We discuss the relationship between these two types of consequences. Specifically, when considering the syntactic class of sentences which contains many of the theorems studied in reverse mathematics (the "Pi-1-2" sentences), any first-order theory has a largest second-order theory associated with it. We show that in many cases, this second-order theory is not axiomatizable. The proof requires new results on models of second-order arithmetic, and in particular suggests a general way of turning results about Peano arithmetic into related results about weaker theories of arithmetic.