- #36
matt grime
Science Advisor
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why are you surprised at the response you got here? we are simply treating the axiom in its own right and not thinking of time. why wuold we think of time? we are attemptinf to make you see that the axiom makes perfect sense and is true for some objects, false for others. if you want to discuss the relevance of the real numbers as a suitalbe description of time, and in particular the fact that the reals satisfy dedekinds axiom and when we discuss time as the use of the reals we get an apparent contradiction, fine. there are many such problems with things like the real numbers or R^3 not least of which is the Banach-Tarski paradox. but that wasn't what you were saying (though it may have been what you meant to say) to our eyes.
there may well be problems if we apply the notion of dekekind's axiom to soemthing physical like time. this doesn't make the axiom problematic in and of itself and is "merely" a question about it's applicability to thinking about it in respect of time. ie he is attempting to describe some physical (or possibly philosophical) problem with the use of the real numbers to describe time. exactly the same could be said for attempting to descrbe lengths using real numbers. see the banach tarski paradox for a well known example. the real numbers are a mathematical invention with some very strange properties, stranger than yoe can ever think of (to paraphrase someone whose name i can't remember). so? that doesn't mean there is a *mathematical* issue with the axiom. the real numbers still satisfy dedekind's axiom, the rationals do not. there are many problems with the real numbers and their suitability to be used in the real world but they are principally of a philosphical nature. indeed there is a reasonable claim that the only numbers we ever need are (a subset of) the constructible ones, and they are countable. so how can something like the real numbers, almost all of which are things we can never know, be suitable for describing something physically meaningful like time? we could just as well be talking abuot the square root of 2 and lengths of hypoteneuses.
an axiom many be problematic (a system of axioms cabn be inconsistent in that they may be mutually contradictory) in that it may be vacuous, always false or just plain silly, and it can be true or false when thought of in relation to some model. the set of dedekind cuts of rationals is a model where the axiom is true. this may make some philosophical problem occur, it may not. i don't think anyone here would care to speculate on what the thoughts of some author we may never have heard of meant in an sentence in some book about time and the relevance of the reals to describe them.
there may well be problems if we apply the notion of dekekind's axiom to soemthing physical like time. this doesn't make the axiom problematic in and of itself and is "merely" a question about it's applicability to thinking about it in respect of time. ie he is attempting to describe some physical (or possibly philosophical) problem with the use of the real numbers to describe time. exactly the same could be said for attempting to descrbe lengths using real numbers. see the banach tarski paradox for a well known example. the real numbers are a mathematical invention with some very strange properties, stranger than yoe can ever think of (to paraphrase someone whose name i can't remember). so? that doesn't mean there is a *mathematical* issue with the axiom. the real numbers still satisfy dedekind's axiom, the rationals do not. there are many problems with the real numbers and their suitability to be used in the real world but they are principally of a philosphical nature. indeed there is a reasonable claim that the only numbers we ever need are (a subset of) the constructible ones, and they are countable. so how can something like the real numbers, almost all of which are things we can never know, be suitable for describing something physically meaningful like time? we could just as well be talking abuot the square root of 2 and lengths of hypoteneuses.
an axiom many be problematic (a system of axioms cabn be inconsistent in that they may be mutually contradictory) in that it may be vacuous, always false or just plain silly, and it can be true or false when thought of in relation to some model. the set of dedekind cuts of rationals is a model where the axiom is true. this may make some philosophical problem occur, it may not. i don't think anyone here would care to speculate on what the thoughts of some author we may never have heard of meant in an sentence in some book about time and the relevance of the reals to describe them.
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