Internal Categoricity in Arithmetic and Set Theory Jouko Väänänen, Tong Wang Abstract: Second order logic was originally considered as an innocuous variant of first order logic in the works of Hilbert. Later study reveals that the analogy with first order logic does not do full justice to second order logic. Quine famously referred to second order logic as "set theory in disguise". Second order logic truly transcends first order logic in terms of strength, and is more appropriate to be compared to (first order) set theory. In second order logic, a large part of set theory becomes essentially logical truth. There is the debate between the "set theory view" and the "second order view" in the foundation of mathematics . The set theory view holds that mathematics is best formalized using first order set theory. The second order view holds that mathematics is best formalized in second order logic. Two important issues in this debate are completeness and categoricity. It is usually conceived that one merit of the set theory view is that first order logic has a complete proof calculus, while second order logic has not. One merit of the second order view is that second order theories of classical structures (e.g. N, R) are categorical, while first order theories allow for non-standard models. The aim of this paper is to synthesize completeness and categoricity in the second order, while working within the framework of normal second order logic instead of full second order logic. Keywords: second order logic; arithmetic; categoricity; set theory