 Let \(A\) be any set of sentences in a first-order language \(\LKe\), where \(K\) includes terminology for arithmetic, and assume that every member of \(A\) is true of the natural numbers. We can even let \(A\) be the set of all sentences in \(\LKe\) that are true of the natural numbers. Then \(A\) has uncountable models, indeed models of any infinite cardinality. Such interpretations are among those that are sometimes called unintended, or non-standard models of arithmetic. Let \(B\) be any set of first-order sentences that are true of the real numbers, and let \(C\) be any first-order axiomatization of set theory. Then if \(B\) and \(C\) are satisfiable (in infinite interpretations), then each of them has denumerably infinite models. That is, any first-order, satisfiable set theory or theory of the real numbers, has (unintended) models the size of the natural numbers. This is despite the fact that a sentence (seemingly) stating that the universe is uncountable is provable in most set-theories. This situation, known as the Skolem paradox, has generated much discussion, but we must refer the reader elsewhere for a sample of it (see the entry on  Skolem’s paradox  and Shapiro 1996).