Skolem's paradox

In mathematical logic, specifically model theory, Skolem's paradox is a direct result of the (downward) Löwenheim-Skolem theorem, which states that every model of a sentence of a first-order language has an elementarily equivalent countable submodel.

Is it a paradox?

The "paradox" is viewed by most logicians as something puzzling, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach-Tarski paradox rather than the sense in Russell's paradox). Timothy Bays has argued in detail that there is nothing in the Löwenheim-Skolem theorem, or even "in the vicinity" of the theorem, that is self-contradictory.

Related Topics:
Banach-Tarski paradox - Russell's paradox - Timothy Bays

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However, some philosophers, notably Hilary Putnam and the Cambridge logician A.W. Moore, have argued that it is in some sense a paradox.

Related Topics:
Hilary Putnam - A.W. Moore

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The difficulty lies in the notion of "relativism" that underlies the theorem. Skolem says:

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: In the axiomatization "set" does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping of M onto Z0 (Zermelo's number sequence). Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a "set" (that is, it does not occur in the domain B).

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Moore (1986) has argued that if such relativism is to be intelligible at all, it has to be understood within a framework that casts it as a straightforward error. This, he argues, is Skolem's Paradox.

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If Skolem's explanation is true, ideas such as countability and uncountability are inherently relative. Our belief that the power set of the natural numbers, P(w), as uncountable, is correct, but must be understood relative to our own current "viewpoint". From another viewpoint this set may in fact be countable. But then it should be possible to make this relativisation explicit. We can do so this only so far as our discourse about sets is intelligible as about a particular collection of objects to which such claims must be relativized. But this in turn is not possible unless we endorse the error that there is a set containing all the sets we mean to talk about.

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"When it is claimed that P(w) is not unconditionally uncountable, we have no way of understanding this except as the demonstrably false claim that it is not uncountable at all."

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We cannot view P(w) from two different points of view at once; that would be incoherent. Nor can we view it simply from this point of view, then the supposed relativity is unintelligible. "But if it were possible to view it from an absolute standpoint, then relativism itself would lose its rationale and there could be no objection to saying that P(w) contained all of w's subsets and that it was unconditionally uncountable."

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