A FORMAL CONSTRUCTION OF THE SPACETIME MANIFOLD

The spacetime manifold, the stage on which physics is played, is constructed ab initio in a formal program that resembles the logicist reconstruction of mathematics. Zermelo’s set theory extended by urelemente serves as a framework, to which physically interpretable proper axioms are added. From this basis, a topology and subsequently a Hausdorff manifold are readily constructed which bear the properties of the known spacetime manifold. The present approach takes worldlines rather than spacetime points to be primitive, having them represented by urelemente. Thereby it is demonstrated that an important part of physics is formally reducible to set theory.     

Axiomatic Quantum Theory

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra–ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities.     

The Simplest Axiom System for Plane Hyperbolic Geometry

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L _B which might be the simplest possible one in that language.     

A Gabbay-Rule Free Axiomatization of T×W Validity

The semantical structures called T×W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, _O, which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator . However, these structures are also suitable for interpreting an extended language, _SO, containing a further possibility operator ^s which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history simultaneity operator. In the present paper we provide an infinite set of axioms in _SO, which is shown to be strongly complete forT ×W-validity. Von Kutschera (1997) contains a finite axiomatization of T×W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions.     

Axiomatizing Relativistic Dynamics without Conservation Postulates

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc ². The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.     

Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains.     

The Basic Algebra of Game Equivalences

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae.The completeness proof is based on reduction of game terms to a certain minimal canonical form, by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities.