  These questions can be made somewhat more concrete by considering a specific case. Let T be a standard, first-order axiomatization of set theory. (For convenience, this entry will focus on the case where T is ZFC, but any standard axiomatization of set theory would do equally well.) On the assumption that T has a model, the Löwenheim-Skolem theorems ensure that it has a countable model. Call this model M. Now, as T  ⊢ ∃x “x is uncountable,” there must be some  mˆ ∈ M such that M  ⊨ “mˆ is uncountable.” But, as M itself is only countable, there are only countably many m ∈ M such that M  ⊨ m ∈  mˆ On the surface, then, we seem to have a straightforward contradiction: from one perspective,  mˆ looks uncountable, while from another perspective,  mˆ is clearly countable.