  Given all this, Skolem's Paradox shows that the line between countable and uncountable sets is, in a fairly deep sense, the first place where our model theory loses the ability to capture cardinality notions. This fact helps to explain why Skolem's Paradox may continue to look paradoxical even after we've absorbed the general points about models and model theory which were presented in the second-to-last paragraph. In short: it's the very fact that we can capture so many of the cardinality notions which live, as it were, below the countable/uncountable distinction which makes our sudden inability to capture the countable/uncountable distinction itself so initially surprising.