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May 15th, 2012, 10:49 PM  #1 
Newbie Joined: May 2012 Posts: 19 Thanks: 0  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 modeltheoretic semantics of these notions on the other. So, in the actual world, which do you think is right  that countability is relative or absolute? 
May 16th, 2012, 05:05 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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...

May 16th, 2012, 04:46 PM  #3 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  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. 
May 16th, 2012, 06:18 PM  #4  
Newbie Joined: May 2012 Posts: 19 Thanks: 0  Re: Skolem's Paradox (a little philosophy) Quote:
 
May 16th, 2012, 06:20 PM  #5 
Newbie Joined: May 2012 Posts: 19 Thanks: 0  Re: Skolem's Paradox (a little philosophy)
maybe not real numbers..., but at least the power set of the natural numbers (to preserve uncountability).

May 16th, 2012, 06:24 PM  #6  
Newbie Joined: May 2012 Posts: 19 Thanks: 0  Re: Skolem's Paradox (a little philosophy) Quote:
 
May 16th, 2012, 08:21 PM  #7 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  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.

May 20th, 2012, 11:48 AM  #8 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  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 LoewenheimSkolem 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!

May 20th, 2012, 05:39 PM  #9  
Newbie Joined: May 2012 Posts: 19 Thanks: 0  Re: Skolem's Paradox (a little philosophy) Quote:
So, the linguistics like higherorder logics as they capture natural language better. The mathematicians like firstorder 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 higherorder 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 nonmodeltheoretic 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 firstorder 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 secondorder logic. He did not say: "Oh, well countability isn't really relative because secondorder logics give us a closer interpretation to our ordinary conception of countability." According to my settheory professor, set theorists usually don't reject the relativity of countability on the grounds of secondorder 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.  
May 21st, 2012, 07:59 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Skolem's Paradox (a little philosophy) Quote:
 

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