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May 15th, 2012, 10:49 PM   #1
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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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May 16th, 2012, 05:05 AM   #2
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Re: Skolem's Paradox (a little philosophy)

I don't think that either concept exists in the "actual world"! There are only 10^80 or so particles in the universe...
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May 16th, 2012, 04:46 PM   #3
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Re: Skolem's Paradox (a little philosophy)

I will not pretend to be an expert in the branches of mathematics most relevant to Skolem's work, but I did take a course in mathematical logic at Penn many a moon ago and have a brother who is quite the mathematical logical sophisticate and from the little I got out of that course and the considerably more I got out of conversations with my brother, my understanding is that Skolem's paradox is not a true paradox even in first order logic and totally disappears as an issue in second order logic. (I should also note that one of the early enthusiasts for second order logic, Rudolph Carnap, had a student later on named Richard Montague, who got interested in natural language semantics and devised what amounts to an arbitrarily high order logic known as Montague Grammar, which revolutionized linguistic semantics in many ways. I took two courses in Montague Grammar in grad school. It was while I was studying for the final in the first of them that I managed to induce the one and only migraine I've ever had! )

So, I'm not sure there is as much to wax philosophical over here as was first believed when Skolem published his result.
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May 16th, 2012, 06:18 PM   #4
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Re: Skolem's Paradox (a little philosophy)

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Originally Posted by CRGreathouse
I don't think that either concept exists in the "actual world"! There are only 10^80 or so particles in the universe...
I could be totally off, but isn't talk of "in the universe" a model with a domain (particles, as you say) and relations between those particles. Since you say "the universe" wouldn't you want to then say that there is an absolute notion of countability: it is the natural extension of what we would ordinarily count as particles. You're counting these particles, and so if we "imagined" that we kept counting, there would an absolute set of natural numbers, real numbers, etc. and an absolute notion of countability and uncountability. Hmm?
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May 16th, 2012, 06:20 PM   #5
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Re: Skolem's Paradox (a little philosophy)

maybe not real numbers..., but at least the power set of the natural numbers (to preserve uncountability).
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May 16th, 2012, 06:24 PM   #6
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Re: Skolem's Paradox (a little philosophy)

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Originally Posted by johnr
my understanding is that Skolem's paradox is not a true paradox even in first order logic and totally disappears as an issue in second order logic.
Agreed. There is no real paradox in mathematics. Timothy Bays's dissertation "Reflections on Skolem's Paradox" goes through almost all of the ways to formulate Skolem's Paradox and shows how each formulation has a "resolution." But, what we learn is that countability is relative. The philosophical question is, "Well, is countability really relative? When we count things, and imagine counting things off into infinity, are we really concerned with relativity? Or are we sure there is a notion of countability?" I think that's how the question goes.
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May 16th, 2012, 08:21 PM   #7
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Re: Skolem's Paradox (a little philosophy)

Well, unfortunately my brother is incommunicado while traveling. So I can't rehash our last conversation with him (that arose completely out of nowhere in a mass email exchange with a large group of people at most one of whom besides me and my bro are likely to have had ANY idea what the discussion was about). But as I reconstruct it, the idea I got from him is that what Skolem really showed was the inadequacy of first order logic in modeling math or set theory. As my brother put it; "It doesn't even get cardinality right." From that I inferred that second order logic DID get cardinality right. But perhaps I read more into his statement than was intended. I will see my brother in a week or so. I'll have to ask him more about this.
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May 20th, 2012, 11:48 AM   #8
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Re: Skolem's Paradox (a little philosophy)

Well, my bro should be in town soon and I hope to have this discussion with him. But it's been illuminating to read around on second order logic and proof theory vs semantics. In linguistics, higher order logic has been viewed as THE great wonder that solves so much that seemed so hopeless in the old days of doing linguistic semantics via translations into first order logic. And model theory is presented as THE cool thing to be doing rather than the old woebegone proof theory. I never really caught the fever, but I saw the point that first order logic really did seem far less than up to the task of capturing natural language semantics. I also took it from Loewenheim-Skolem that the problem was the inadequacy of first order logic. So it was very eye opening to read around and see how much scorn there is in mathematical logic for higher order logics precisely BECAUSE of the loss of a complete proof theory of the sort we know and love from first order logic. So it looks like there may be more of a philosophical discussion to be had here than I thought!
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May 20th, 2012, 05:39 PM   #9
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Re: Skolem's Paradox (a little philosophy)

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Originally Posted by johnr
I also took it from Loewenheim-Skolem that the problem was the inadequacy of first order logic. So it was very eye opening to read around and see how much scorn there is in mathematical logic for higher order logics precisely BECAUSE of the loss of a complete proof theory of the sort we know and love from first order logic. So it looks like there may be more of a philosophical discussion to be had here than I thought!
Haha..finally I don't feel so crazy. Asking the philosophical question around has only gotten me brusque replies.

So, the linguistics like higher-order logics as they capture natural language better. The mathematicians like first-order logics as we enjoy nice proofs. Is this where you see the philosophical question? I'm sure there is one here, but I might be asking a different one.

When we ask the question, "Does first- or higher-order logics capture 'countability' better?" I am asking what the "perspective" we are even trying to capture says about "countability." What does our ordinary conception of countability tell us? Do we ordinarily take countability to be relative? It seems not as we will call models countable -- and certainly assuming ZFC, models can't say of themselves that they are countable. So, from our "outside" perspective, or non-model-theoretic perspective, what we we take countability to be? An absolute or relative notion? I think we assume that its absolute.

But this brings up more (ontological) questions: is countability just a notion or concept, or does it have an instantiation in nature? CRGreathouse brusquely replied that there can't be an instantiation of a countable set, or something countable, because our universe is merely finite. I agree, but what do we mean then when we ordinarily take countability to be absolute?

Some more rambling: if our ordinary notion of countability was relative, what would it be relative to? In first-order logics, A's being countable depends on what is in the model -- as I described in the OP. I don't know much about second order logic. But, I do know that Skolem rejected second-order logic. He did not say: "Oh, well countability isn't really relative because second-order logics give us a closer interpretation to our ordinary conception of countability." According to my set-theory professor, set theorists usually don't reject the relativity of countability on the grounds of second-order logic, but on philosophical grounds. For example, we can count coconuts, if there were an infinite amount of them, why should we think that the countability of the coconuts should be relative? As I replied to CRGreathouse, maybe it is because we take the universe to be a fixed domain; that is, there is really only one model, and therefore, only one (absolute and fixed) notion of countability.
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May 21st, 2012, 07:59 PM   #10
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Re: Skolem's Paradox (a little philosophy)

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Originally Posted by sunvogue
CRGreathouse brusquely replied that there can't be an instantiation of a countable set, or something countable, because our universe is merely finite.
Yeah, the ordinary universe is pretty boring.
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