Set Theoretical Approach for Special Relativity
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From: Nam Nguyen on 2 Nov 2009 01:58 In the thread "Counterintuitions and the well-ordering theorem" (sci.logic) I briefly speculated about a set-theoretical approach for SR (Special Relativity). In this thread a little more details on this "relativistic" set formal system will be presented. Such a formal system would require infinite numbers of epsilon relation symbols e1, e2, en, ... But for the time being we'd concentrate on a system with only 2 symbols e1 and e2 and would come back to the general case later. *** First we'll formulate the system, say 2ZFC, and then some motivations on the formulation will be presented. The axioms of 2ZFC, written in L(e1, e2), would fall into 3 following groups: (1) - ZFC Axioms: - A set of axioms that has formulas using only e1 which all together would form ZFC axioms w.r.t. e1. - A set of axioms that has formulas using only e2 which all together would form ZFC axioms w.r.t. e2. (2) - Frame-of-Reference Axioms: - Axy[(Au[~(ue1x)] /\ Av[~(ve2y)]) -> (x=y)] (3) - Axioms of Relativity: - ExEy[ Az[ze1x <-> ze1y] <-> ~(Az[ze2x <-> ze2y])] /\ Az[ze2x <-> ze2y] <-> ~(Az[ze1x <-> ze1y])] ] *** Motivations for the Axioms (2) and (3). ====================================== Two facts that seem to often have escaped attention when talking about SR. Firstly, the particular value 300,000 km/s of the constant speed of light is immaterial to SR: it's the constancy - not the value - that matters. But consequently, in an abstract sense, each constant value would characterize a _distinct context for relativity_. For intance, suppose there exists 2 physical universes that have no physical connections whatsoever, one of which the constant speed of light is 300,000 km/s, while the other it's twice as slow, 150,000 km/s. In such a context, we can not just refer to 2 frames of reference, without mentioning which speed of light - hence which universe - is the underlying context to discuss SR. In the same sense, the axioms (only 1 axiom for the case of 2 epsilon relations, for now) of group (2) would establish a context, or a framework of relativity. That context is the empty set being identical as stipulated by the axiom in this group (2). Secondly, it's not true all truths about SR is relative. The constancy of the speed of light is an absolute truth. Similarly, the axiom in (2) is one of the "absolute" truths since it establishes the constancy of the existence of the empty set. The axioms in group (3) would model after Lorentz (space-time) transformations between 2 frames of reference: http://en.wikipedia.org/wiki/Relativity_of_simultaneity Note that the formulas in these transformations are interchangeable between the 2 corresponding observers. This interchangeability is the basis of the axioms in group (3). --- But if course this is only a preliminary details. My hope is that these details wouldn't too wrong. But we'll see I guess. |