Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 6 / c

Skolem's paradox is the apparent conflict between a countable model of a sufficiently strong set theory and that model's theorem asserting the existence of uncountable sets. Downward Löwenheim-Skolem theorem gives a countable elementary model whenever an appropriate infinite model exists; internally the same model still proves, for example, that its real-number set is uncountable.

The two uses of “countable” quantify over different collections of functions. Externally the whole model, and therefore the elements it regards as members of its real-number set, can be enumerated. Internally a set is countable only if an enumerating function exists among the sets of the model. The external enumeration need not be an element of the model.

Thus there is no contradiction: the model's assertion is that no function it contains is the required enumeration. Cantor's theorem remains valid inside the model. The model need not be transitive or have the actual standard natural numbers for this explanation; the essential issue is the restriction of internal quantifiers to its own domain.