Implicit Definability in Arithmetic

Implicit Definability in Arithmetic

We consider implicit definability over the natural number system N,+,×,=. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of N which are not explicitly definable from each other. The second theorem says that there exists a subset of N which is not implicitly definable but belongs to a countable, explicitly definable ...

Existential Definability in Arithmetic

Definability and Decision Problems in Arithmetic

Introduction. In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems. In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of s...

Structure and Definability in General Bounded Arithmetic Theories

The bounded arithmetic theories Ri 2, S i 2, and T i 2 are closely connected with complexity theory. This paper is motivated by the questions: what are the Σi+1-definable multifunctions of R i 2? and when is one theory conservative over another? To answer these questions we consider theories R̂i 2, Ŝ i 2, and T̂ i 2 where induction is restricted to prenex formulas. We also define T̂ i,τ 2 which ha...

Implicit definability of truth constants in Łukasiewicz logic

In the framework of propositional Lukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly defining each of the rational elements in the standard semantics are explored, and based on that, a faithful interpretation of theories in Rational Pavelka logic ...