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.