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The main contribution of this paper is a formal characterization of recursive object specifications and their existence based on a denotational untyped semantics of the object calculus. Existence is not guaranteed but can be shown employing Pitts' results on relational properties of domains. The semantics can be used to analyse and verify Abadi and Leino's object logic but it also suggests extensions. For example, specifications of methods may not only refer to fields but also to methods of objects in the store. This can be achieved without compromising the existence theorem. An informal logic of predomains is in use intentionally in order to avoid any commitment to a particular syntax of specification logic.