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I guess there are two questions here.


QUESTION 1: Skolem's Paradox shows that countability is relative in first-order logic, but where is the relativity? In this first question, I will do the following: (A) I will give an example here of the relativity of countability and then (B) talk about why I don't see the relativity. I am certain that what I write in (A) is right, but I am equally certain that I am wrong about what I write in (B). My question is where I am going wrong in (B). However, this might be a mathematical question, and almost certainly stems from my minimal understanding of set theory (although, I have been working for weeks trying to understand it).

(A) An example of the relativity of countability in first-order logic: Let A be an uncountable set in (a model) M. This means there is no bijection f in M between A and the set of naturals w. Now, add f to M and let this union be a new model N. Therefore, A is countable in N. So, whether a set A is countable or not is relative to what bijections with A some model recognizes.

(B) However, in order to talk about the model theory of set theory, we have to assume that set theory is consistent. Therefore, we must assume that there is some V (universe of sets) that serves as the model of set theory. Let's just use ZFC. Thus all the model theory we do on ZFC is done in V. That is, all models of ZFC that we look at are subsets of V. Now, it seems to me that this implies that there is an absolute perspective -- an ultimate arbiter of whether or not some set is countable. If A is countable according to V, then A is absolutely countable.

Indeed, some model M might not recognize a bijection between A and the set of naturals while V does (and thus A is absolutely countable), or M might also get the set of naturals wrong (i.e., it is not the same set that V sees as the set of naturals). Thus we can say from an absolute perspective whether a set A is (un)countable, whether some model is (un)countable, and we can say whether or not some model recognizes the right bijections or sets (e.g. is the set of naturals in M the same as V's?). And so, countability is not relative -- there is an absolute perspective, viz.: V's perspective.

Given (A) there is indeed a sense in which countability is relative, but it is (for lack of a better term) a local relativity: two models may differ on what they take to be w or f. But, this doesn't mean that countability is absolutely relative; again, isn't V the absolute perspective?

12 June 2012 EDIT: I asked about this at math.SE and the answer is simply that we don't just work in the universe. Sets that exist in the universe that are not in some model M under consideration are irrelevant for model theory in M -- at least, that's what I gathered.


QUESTION 2: Skolem's Paradox is not at all a problem for mathematics -- that is, no contradiction can be derived from (LS-Theorem ∧ Cantor's Theorem).

However, philosophers (I am told) still find the "paradox" interesting; they for some reasons find Skolem's Paradox still paradoxical, despite its not being a problem for mathematics. I ask: what are these reasons? why is it still paradoxical? why is it still philosophically interesting?

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A Google search for Skolem and Badiou (an important contemporary philosopher-mathematician) turned up this Livingston paper which might be worth a look. Skolem and this paradox show up in section 3, pp. 16-18; here's an excerpt of some of the theoretical motivation for the concern of philosophers (especially of mathematics and logic):

In 1977, Hilary Putnam delivered to the Association for Symbolic Logic an address entitled “Models and Reality.” In it, he considers the status of models and model-theoretic reasoning in order to illuminate the larger metaphysical question of the bearing of rational thought and language on the world. Putnam begins by considering a familiar result in model theory, the Löwenheim-Skolem theorem, which establishes that every abstract theory which has any infinite model (of any cardinality, no matter how large) also has an infinite model of the very first (or smallest) cardinal size, the size of the “countable” set of natural numbers, which is symbolized as א"0”. This result leads directly to a somewhat counter-intuitive implication that has sometimes been termed “Skolem‟s paradox”: the paradox (or seeming one) is that any statement about transfinite sets and their cardinalities (no matter how large) can be re-interpreted in a countable model, and so can be modeled by structures of (only) countable size. It follows that any arbitrary statement about transfinite cardinalities – for instance the statement that there is at least one non-countable set – can be re-interpreted in a model with only countable sets and so can apparently hold true in a model that “actually” falsifies it. The usual way of handling this paradox within mathematical set theory, suggested already by Skolem himself, is to point out that the plurality of possible models means that cardinality is itself a “relative” notion. In particular, since the cardinality of a set is defined by the possibilities of its being put (or not) into one-to-one correspondence with other sets within the same model, the apparent “collapse” of cardinality in the countable models guaranteed to exist by the Löwenheim-Skolem theorem is simply a consequence of the availability, in certain models, of relations that are not available in others. A statement involving the existence of a nondenumerable set is then seen as true “in reality” even though it can be verified by a wholly countable model, owing simply to the relative lack of relations in that model. Moreover, it is clear (as the usual gloss on Skolem‟s paradox emphasizes) that the countable model cannot, here, be the “intended” one, which is after all supposed to be the whole universe of sets, not just some limited, countable ersatz.

A footnote indicates that Putnam's presentation (referred to in the first sentence of my cite) was reprinted as Putnam, H. “Models and Reality,” The Journal of Symbolic Logic, Vol. 45, No. 3 (Sep., 1980), pp. 464-482.

Livingston goes on to suggest that the "standard response" to the paradox "defuses" it of some of its very real paradoxicality. At any rate I'd definitely recommend taking at least a quick look through the article (especially the section on Skolem); I might also suggest reading a bit about Badiou, who you may find interesting in his own right -- and please feel free to share questions on any further concerns that may come up in your study. (Welcome to Philosophy.SE, by the way!)

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    The most common response I hear these days, and the one I'm inclined towards myself, is that what Lowenheim-Skolem shows us is something about the limits of formalization. In particular, it show us that first-order logic is inadequate for formalizing mathematical theories (the theorem doesn't hold in a second-order setting). This has pushed many to adopt second-order formulations of, e.g., Peano Arithmetic despite supposed worries about the ontological status of Second-Order Logic. – Dennis Jan 27 '13 at 4:14
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Here's how I think of it. We mathematicians are working from within some V, which evidently satisfies the axioms of ZFC. From our point of view there is no surjection from w, the natural numbers, onto P(w), the power set of the natural numbers. However, all we're really proving is that there's no surjection within our universe, V. But if V is really just a sub-universe of a bigger universe V', then such a surjection might exist in V' and it might be obvious to people working from within V' that P(w) is countable. The point is that it's impossible to distinguish whether we're working in "the universe" (the next paragraph shows that even such a concept might be invalid) or a submodel thereof.

Regarding "Therefore, we must assume that there is some V (universe of sets) that serves as the model of set theory. Let's just use ZFC. Thus all the model theory we do on ZFC is done in V. That is, all models of ZFC that we look at are subsets of V. Now, it seems to me that this implies that there is an absolute perspective -- an ultimate arbiter of whether or not some set is countable." There isn't necessarily a single such V.

The Stanford Encyclopedia of Philosophy (SEP) has a nice article on Skolem's paradox and Tim Bays did his dissertation on it (and that dissertation is heavily cited in the SEP article). I think the SEP article and Bays' thesis have pretty much dispelled all the paradoxicality of Skolem's paradox.

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In response to your first question, I think you are almost correct given a slight modification: if you redefine your notions of countable to be

Given a model M satisfying ZF, a set A in M is M-countable iff there exists a bijection between A and the natural numbers.

A is universally ZF-countable iff for all M satisfying ZF and containing A, there exists (in that M) a bijection between A and the natural numbers.

A is existentially ZF-countable iff there exists M satisfying ZF and containing A wherein there exists a bijection between A and the natural numbers

then you can recover universality (i.e. all valid models agree, or some valid model agrees). This is a move which is generally valid in both mathematics and philosophy: equating that which is logically necessary (or possible) with that which is universal (or exists).

Adding additional axioms (like the axiom of choice) is not okay because you're making a different statement: if something is ZFC-countable then it has little bearing on the properties of sets in a model that does not obey the axiom of choice. You are essentially saying, "I don't like those models; let us not speak of them," and throwing away the examples of non-universality in order to obtain apparent universality!

I do not know, however, what if any work has been done on universal/existential-ZF-countability. I also do not know why this ought to be a philosophically interesting question (beyond the thought it takes to notice that there may be different notions of countability-in-M and universal/existential-ZF-countability). The actual world is not just countable but finite, and our logical frameworks have no trouble extending to uncountable systems, so this observation seems to have little bearing on other affairs.

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