= Solution
<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 \b[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.
Back to article page
