Title: Comparing Classical and Relativistic Kinematics in First-Order Logic,
Abstract: The aim of this paper is to present a new logic-based understanding of the
connection between classical kinematics and relativistic kinematics. We show
that the axioms of special relativity can be interpreted in the language of
classical kinematics. This means that there is a logical translation function
from the language of special relativity to the language of classical kinematics
which translates the axioms of special relativity into consequences of
classical kinematics. We will also show that if we distinguish a class of
observers (representing observers stationary with respect to the "Ether") in
special relativity and exclude the non-slower-than light observers from
classical kinematics by an extra axiom, then the two theories become
definitionally equivalent (i.e., they become equivalent theories in the sense
as the theory of lattices as algebraic structures is the same as the theory of
lattices as partially ordered sets). Furthermore, we show that classical
kinematics is definitionally equivalent to classical kinematics with only
slower-than-light inertial observers, and hence by transitivity of definitional
equivalence that special relativity theory extended with "Ether" is
definitionally equivalent to classical kinematics. So within an axiomatic
framework of mathematical logic, we explicitly show that the transition from
classical kinematics to relativistic kinematics is the knowledge acquisition
that there is no "Ether", accompanied by a redefinition of the concepts of time
and space.