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Math Help - Skolem's Paradox (a little philosophy)

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    Skolem's Paradox (a little philosophy)

    Skolem "showed" that the notion of countability is relative.

    In set theory, if there is a set consisting of a bijection between a set A and the set of naturals N in the domain some model M, then A is countable in M. Now, suppose that we removed the bijections between A and N from M and called this new model M'. According to M', A is uncountable. And so, Skolem "shows" that there is no absolute notion of a set being countable.

    However, we ordinarily talk as if there is an absolute notion; and indeed, we wouldn't be struck by Skolem's paradox if we didn't ordinarily treat countability as an absolute notion. One of the things that Skolem's Paradox reveals is the gap between the ordinary English semantics of countability, sets, quantification, membership, etc. on the one hand and the model-theoretic semantics of these notions on the other.

    So, in the actual world, which do you think is right -- that countability is relative or absolute?
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    Re: Skolem's Paradox (a little philosophy)

    i don't think we KNOW. in fact, the possibility of there being "uncountable models" and "countable models" highlights the difficulty of "talking about real things". the very encoding of information in language, limits us to something that is somewhat less than "what there is", so that we can describe it without describing it "fully" (which would be pointless, since the universe has already "described itself" better than we could).

    i think that, at best, we "hope" the models we come up with are appropriate, insofar as they don't contradict our experience. there is nothing about the physical world that suggests it is even infinite, much less uncountably so. but our experience with things that appear to "vary continually" suggests that describing the universe (or some bits of it) as a continuum is USEFUL. at some point, the formalism itself takes over, and research is carried on of questions that may never "apply" to the real world, but are "interesting". and why not? a painting does not have to have "photo-realism" to be meaningful to those who view it.

    i suspect the relative view is more accurate: i doubt ANY formal system will ever "capture the known universe". i believe that we need the freedom to look at things under different (and perhaps even contradictory) lenses to get "the whole picture". these are, however, merely my own views, and many people disagree.
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    Re: Skolem's Paradox (a little philosophy)

    Quote Originally Posted by Deveno View Post
    i think that, at best, we "hope" the models we come up with are appropriate, insofar as they don't contradict our experience. there is nothing about the physical world that suggests it is even infinite, much less uncountably so. but our experience with things that appear to "vary continually" suggests that describing the universe (or some bits of it) as a continuum is USEFUL.
    Just to see if I'm reading you right: would an example here be (the folkloric) Newton working out calculus to try to explain the falling apple? Or just Newtonian physics? It's a model that helps us "describe" the universe, and we "hope" that it corresponds to the universe's "description of itself." Then Einstein comes along and says, "Sorry Newton, your model does not so correspond, but mine does." And then the Quantum physicists come in and say, "Sorry Einstein, we're working on something that conflicts with your theory." But, we "hope" that this is a progressive dialectic, one that gets us closer to the description.
    Quote Originally Posted by Deveno View Post
    at some point, the formalism itself takes over, and research is carried on of questions that may never "apply" to the real world, but are "interesting". and why not? a painting does not have to have "photo-realism" to be meaningful to those who view it.
    I agree. But, here you're saying something different than before, right? It's like applied math verses pure math. This reminds me of Cantor's rant in the Grandlagen about the freedom of mathematics. (Pure) Math is free in that you go where it leads; you just start with some statements and see what follows, without regard for the metaphysical implications. On the other hand, Einstein refused to accept the quantum theories because they defied local realism.
    Quote Originally Posted by Deveno View Post
    i suspect the relative view is more accurate: i doubt ANY formal system will ever "capture the known universe". i believe that we need the freedom to look at things under different (and perhaps even contradictory) lenses to get "the whole picture". these are, however, merely my own views, and many people disagree.
    I might be reading this wrong, but I think you're disagreeing with yourself. When you talk about capturing "the known universe" or talking about "the whole picture" aren't you talking about something absolute? Indeed, it might be useful or mathematically aesthetic (whatever that is...I just made it up) for their to be relativity and (you seem to think, therefore...) freedom. But, in the end, in the final description, the description the universe "gives of itself" isn't there now no relativity?
    Last edited by mpitluk; May 16th 2012 at 06:46 PM.
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