where T is a theory and each Ai, 1 ≤ i ≤ n, is a formula (an axiom of T). The expression ‘∃f A1…An’ is to be read as ‘there are finitely many formulas A1…An’. (This quantifier is not first-order. However, I am not going to press the point that the nominalist seems to need a non-first-order quantifier to express a property typical of first-order logic. This is only one of the worries we are leaving aside in this formulation.) This version is parasitic on the following platonistic formulation of the compactness theorem: