Axiomatic set theory ends in meaninglessness

no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur

Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

he "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.

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

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

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).

Moore (1985) 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.

Zermelo at first declared the Skolem paradox a hoax. In 1937 he wrote a small note entitled "Relativism in Set Theory and the So-Called Theorem of Skolem" in which he gives (what he considered to be) a refutation of "Skolem's paradox", i.e. the fact that Zermelo-Fraenkel set theory --guaranteeing the existence of uncountably many sets-- has a countable model. His response relied, however, on his understanding of the foundations of set theory as essentially second-order (in particular, on interpreting his axiom of separation as guaranteeing not merely the existence of first-order definable subsets, but also arbitrary unions of such). Skolem's result applies only to the first-order interpretation of Zermelo-Fraenkel set theory, but Zermelo considered this first-order interpretation to be flawed and fraught with "finitary prejudice". Other authorities on set theory were more sympathetic to the first-order interpretation, but still found Skolem's result astounding:

* At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known. (John von Neumann)

* Skolem's work implies "no categorical axiomatisation of set theory (hence geometry, arithmetic [and any other theory with a set-theoretic model]...) seems to exist at all". (John von Neumann)

* Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached. (Abraham Fraenkel)

* I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique. (Skolem)

Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it

no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur

One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction

The good news is that this strongly paradoxical reading is optional. The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading is to insist that the real numbers have some elusive, essential property not captured by system S. This view is usually associated with a Platonism that permits its proponents to say that the real numbers have certain properties independently of what we are able to say or prove about them.

The problem with this view is that LST proves that if some new and improved S' had a model, then it too would have a countable model. Hence, no matter what improvements we introduce, either S' has no model or it does not escape the air of paradox created by LST. (S' would at least have its own typographical expression as a model, which is countable.)

Finally, there is the working faith of the working mathematician whose specialization is far from model theory. For most mathematicians, whether they are Platonists or not, the real numbers are unquestionably uncountable and the limitations on formal systems, if any, don't matter very much. When this view is made precise, it probably reduces to the second view above that LST proves an unexpected limitation on formalization. But the point is that for many working mathematicians it need not, and is not, made precise. The Skolem paradox has no sting because it affects a "different branch" of mathematics, even for mathematicians whose daily rounds take them deeply into the real number continuum, or through files and files of bytes, whose intended interpretation is confidently supposed to be univocal at best, and at worst isomorphic with all its fellow interpretations.