In this paper, we propose a variant of Answer Set Programming (ASP) with\nevaluable functions that extends their application to sets of objects,\nsomething that allows a fully logical treatment of aggregates. Formally, we\nstart from the syntax of First Order Logic with equality and the semantics of\nQuantified Equilibrium Logic with evaluable functions (QELF). Then, we proceed\nto incorporate a new kind of logical term, intensional set (a construct\ncommonly used to denote the set of objects characterised by a given formula),\nand to extend QELF semantics for this new type of expression. In our extended\napproach, intensional sets can be arbitrarily used as predicate or function\narguments or even nested inside other intensional sets, just as regular\nfirst-order logical terms. As a result, aggregates can be naturally formed by\nthe application of some evaluable function (count, sum, maximum, etc) to a set\nof objects expressed as an intensional set. This approach has several\nadvantages. First, while other semantics for aggregates depend on some\nsyntactic transformation (either via a reduct or a formula translation), the\nQELF interpretation treats them as regular evaluable functions, providing a\ncompositional semantics and avoiding any kind of syntactic restriction. Second,\naggregates can be explicitly defined now within the logical language by the\nsimple addition of formulas that fix their meaning in terms of multiple\napplications of some (commutative and associative) binary operation. For\ninstance, we can use recursive rules to define sum in terms of integer\naddition. Last, but not least, we prove that the semantics we obtain for\naggregates coincides with the one defined by Gelfond and Zhang for the Alog\nlanguage, when we restrict to that syntactic fragment. (Under consideration for\nacceptance in TPLP)\n