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General relativistic hypercomputing and foundation of mathematics
"... Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, ..."
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Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert’s programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential nonTuring computations. Additionally, new “engineering ” ideas will be outlined for solving the socalled blueshift problem of GRcomputing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome DurandLose, Mark Hogarth and Martin Ziegler, will be indicated. 1
Mathematical logic for life science ontologies
 DE QUEIROZ (EDS.), LOGIC, LANGUAGE, INFORMATION AND COMPUTATION, 16TH INT. WORKSHOP, WOLLIC 2009
, 2009
"... We discuss how concepts and methods introduced in mathematical logic can be used to support the engineering and deployment of life science ontologies. The required applications of mathematical logic are not straighforward and we argue that such ontologies provide a new and rich family of logical th ..."
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We discuss how concepts and methods introduced in mathematical logic can be used to support the engineering and deployment of life science ontologies. The required applications of mathematical logic are not straighforward and we argue that such ontologies provide a new and rich family of logical theories that wait to be explored by logicians.
Modelling Quantum Theoretical Trajectories within Geometric Relativistic Theories
, 2009
"... Andréka and her colleagues have described various geometrically inspired firstorder theories of special and general relativity, while Székely’s PhD dissertation focuses on an intermediate logic of accelerated observers. Taken together, these theories provide an impressive foundation on which to bui ..."
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Andréka and her colleagues have described various geometrically inspired firstorder theories of special and general relativity, while Székely’s PhD dissertation focuses on an intermediate logic of accelerated observers. Taken together, these theories provide an impressive foundation on which to build wider mathematical descriptions of physical reality, but they remain deficient in one important respect — they do not include direct support for quantum theory. In this paper we will attempt to remedy this situation by incorporating a model of quantum theoretical trajectories that can reasonably claim to be physically meaningful. We have recently shown that the ‘bidirectional model ’ of quantum trajectories — in which particles are deemed to ‘hop ’ randomly from one spacetime location, q, to another, q ′ (which can be either earlier or later in time than q), and in which paths comprise a finite number of hops — is logically equivalent to Feynman’s pathintegral formulation when spacetime is assumed to be Euclidean. In this paper we extend the model to relativistic spacetimes, and argue that observers are subject to the same ‘quantum illusions ’ as in the Euclidean case — for, even though motion is discrete and respects no builtin ‘arrow of time’, observers have no choice but to perceive particle trajectories as continuous (locally) futurepointing paths in spacetime. Whereas the relativistic theories presuppose continuous paths as part of their axioms, the ‘quantum illusion ’ of continuous motion allows us to replace this axiom with a lowerlevel quantuminspired axiom concerning discrete jumps in spacetime. We investigate the nature of these jumps, and the extent to which they can be tied to the underlying geometric structure of spacetime. In particular, we consider hops of the form q → q ′ which preserve features of the underlying number field, and investigate the extent to which all hops can be restricted to be of this form. 1
COMPARING RELATIVISTIC AND NEWTONIAN DYNAMICS IN FIRSTORDER LOGIC
"... In this paper we introduce and compare Newtonian and relativistic dynamics as two theories of firstorder logic (FOL). To illustrate the similarities between Newtonian and relativistic dynamics, we axiomatize them such that they differ in one axiom only. This one axiom ..."
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In this paper we introduce and compare Newtonian and relativistic dynamics as two theories of firstorder logic (FOL). To illustrate the similarities between Newtonian and relativistic dynamics, we axiomatize them such that they differ in one axiom only. This one axiom
Vienna Circle and Logical Analysis of Relativity Theory
, 2009
"... 1 introduction In this paper we present some of our school’s results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain firstorder logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main a ..."
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1 introduction In this paper we present some of our school’s results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain firstorder logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main aims of our school are the following: We want to base the theory on simple, unambiguous axioms with clear meanings. It should be absolutely understandable for any reader what the axioms say and the reader can decide about each axiom whether he likes it. The theory should be built up from these axioms in a straightforward, logical manner. We want to provide an analysis of the logical structure of the theory. We investigate which axioms are needed for which predictions of RT. We want to make RT more transparent logically, easier to understand, easier to change, modular, and easier to teach. We want to obtain deeper understanding of RT. Our work can be considered as a casestudy showing that the Vienna