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On Tarski's axiomatic foundations of the calculus of relations
It is shown that Tarski's set of ten axioms for the calculus of relations is
independent in the sense that no axiom can be derived from the remaining
axioms. It is also shown that by modifying one of Tarski's axioms slightly, and
in fact by replacing the right-hand distributive law for relative
multiplication with its left-hand version, we arrive at an equivalent set of
axioms which is redundant in the sense that one of the axioms, namely the
second involution law, is derivable from the other axioms. The set of remaining
axioms is independent. Finally, it is shown that if both the left-hand and
right-hand distributive laws for relative multiplication are included in the
set of axioms, then two of Tarski's other axioms become redundant, namely the
second involution law and the distributive law for converse. The set of
remaining axioms is independent and equivalent to Tarski's axiom system
A further simplification of Tarski's axioms of geometry
A slight modification to one of Tarski's axioms of plane Euclidean geometry
is proposed. This modification allows another of the axioms to be omitted from
the set of axioms and proven as a theorem. This change to the system of axioms
simplifies the system as a whole, without sacrificing the useful modularity of
some of its axioms. The new system is shown to possess all of the known
independence properties of the system on which it was based; in addition,
another of the axioms is shown to be independent in the new system.Comment: 10 page
Is there life beyond Quantum Mechanics?
We formulate physically-motivated axioms for a physical theory which for
systems with a finite number of degrees of freedom uniquely lead to Quantum
Mechanics as the only nontrivial consistent theory. Complex numbers and the
existence of the Planck constant common to all systems arise naturally in this
approach. The axioms are divided into two groups covering kinematics and basic
measurement theory respectively. We show that even if the second group of
axioms is dropped, there are no deformations of Quantum Mechanics which
preserve the kinematic axioms. Thus any theory going beyond Quantum Mechanics
must represent a radical departure from the usual a priori assumptions about
the laws of Nature.Comment: 23 pages, latex. v3: commentaries on the axioms expanded, a
non-technical summary added, references added, typos fixed. v4: version
accepted for publication in Journal of Mathematical Physic (under a different
title). Axiomatics is simplified and the number of axioms reduced, some
proofs clarified, typos fixe
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