Skolem paradox

A consequence of the Löwenheim–Skolem theorem (see Gödel completeness theorem), stating that every consistent formal axiomatic theory defined by a countable family of axioms has a countable model. In particular, if one assumes the consistency of the axiom system of the Zermelo–Fraenkel set theory or of the elementary theory of types (see Axiomatic set theory), then there is a model (cf. Interpretation) of these theories with countable domain. This is so despite the fact that these theories were designed to describe very extensive fragments of naïve set theory, and within the limits of these theories one can prove the existence of sets of very large uncountable cardinality, so that in any model of these theories there must exist uncountable sets.

It must be stressed that Skolem's paradox is not a paradox in the strict sense of the word, that is, in no way does it show the inconsistency of the theory within whose limits it is established (see also Antinomy). For example, in a countable model of Zermelo–Fraenkel theory, every set is countable from an external point of view. However, in set theory the existence of uncountable sets is provable; so the model also contains sets $S$ which are uncountable from an internal point of view, in the sense that inside the model there is no enumeration of the set $S$.

References

[1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)