Comparing Relativistic and Newtonian Dynamics in First-Order Logic
In this paper we introduce and compare Newtonian and relativistic dynamics as two theories of first-order logic (FOL). To illustrate the similarities between Newtonian and relativistic dynamics, we axiomatize them such that they differ in one axiom only.
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Comparing Relativistic and Newtonian Dynamics in First-Order Logic Introduction In this paper we introduce and compare Newtonian and relativistic dynamics as two theories of first-order 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 difference, however, leads to radical differences in the predictions of the two theories. One of their major differences manifests itself in the relation between relativistic and rest masses, see Thms. 5 and 6. The statement that the center-lines of a system of point masses viewed from two different reference frames are related exactly by the coordinate transformation between them seems to be a natural and harmless assumption; and it is natural and harmless in Newtonian dynamics, see Cor. 11. However, in relativistic dynamics it leads to a contradiction, see Thm. 4. Showing this surprising fact, which also illustrates the great difference between the two theories, is the main result of this paper. Our work is directly related to Hilbert’s 6th problem on axiomatization of physics. Moreover, it goes beyond this program since our general aim is not only to axiomatize physical theories but to investigate the relationship between the basic assumptions (axioms) and the predictions (theorems) of the theories and to compare the axiom systems of related theories. Our another general aim is to provide a foundation of physics similar to that of mathematics. For good reasons, the foundation of mathematics was performed strictly within FOL. One of these reasons is that staying within FOL helps to avoid tacit assumptions. Another reason is that FOL has a complete inference system while second-order logic (and thus any higher-order logic) cannot have one, see, e.g., (11, §IX. 1.6). For further reasons for staying within FOL, see, e.g., (5, §Why FOL?), (1), (35, §11), (19), (20). A. Máté et al. (eds.), Der Wiener Kreis in Ungarn / The Vienna Circle in Hungary © Springer-Verlag/Wien 2011
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Judit X. Madar´ asz and Gergely Sz´ekely
There are many FOL axiomatizations of relativistic kinematics both special and general, see, e.g., (1), (8), (9), (16), (30). However, as far as we know, our co-authored paper (6) is the only one which deals with the FOL axiomatization of relativistic dynamics, too. Newtonian and relativistic kinematics are compared in the level of axioms in (5, §4.1). The main aim of this paper is to compare the key axioms and theorems of Newtonian and relativistic dynamics, too.
A first-order logic frame for dynamics Our choice of vocabulary (basic concepts) is explained as follows. We represent motion as the changing of spatial location of bodies in time. To do so, we have reference-frames for coordinatizing events (sets of bodies) and, for simplicity, we associate reference-frames with observers. There are special kind of bodies which we call photons. For coordinatizing events, we use an ordered field in place of the field of real numbers.1 Thus the elements of this field are the q
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