The theory set out in this book is the result of the confluence and common development of two currents of mathematical research, Descriptive Set Theory and Recursion Theory. Both are concerned with notions of definability and, more broadly, with the classification of mathematical objects according to various measures of their complexity. These are the common themes which run through the topics discussed in this book.

Descriptive Set Theory arose around the turn of the century as a reaction among some of the mathematical analysts of the day against the free-wheeling methods of Cantorian set theory. People such as Baire, Borel, and Lebesgue felt uneasy with constructions which required the Axiom of Choice or the set of all countable ordinals and began to investigate what part of analysis could be carried out by more explicit and constructive means. Needless to say, there was vigorous disagreement over the meaning of these terms. Some of the landmarks in the early days of Descriptive Set Theory are the construction of the Borel sets, Suslin's Theorem that a set of real numbers is Borel just in case both it and its complement are analytic, and the discovery that analytic sets have many pleasant properties — they are Lebesgue measurable, have the Baire property, and satisfy the Continuum Hypothesis.

A natural concomitant of this interest in the means necessary to effect mathematical constructions is the notion of hierarchy. Roughly speaking, a hierarchy is a classification of a collection of mathematical objects into levels, usually indexed by ordinal numbers. Objects appearing in levels indexed by larger ordinals are in some way more complex that those at lower levels and the index of the first level at which an object appears is thus a measure of the complexity of the object. Such a classification serves both to deepen our understanding of the objects classified and as a valuable technical tool for establishing their properties.

A familiar example, and one which was an important model in the development of the theory, is the hierarchy of Borel sets of real numbers. This class is most simply characterized as the smallest class of sets containing all intervals and closed under the operations of complementation and countable union.

Gödel's version of the modal ontological argument for the existence of God has been criticized by J. Howard Sobel [5] and modified by C. Anthony Anderson [1]. In the present paper we consider the extent to which Anderson's emendation is defeated by the type of objection first offered by the Monk Gaunilo to St. Anselm's original Ontological Argument. And we try to push the analysis of this Gödelian argument a bit further to bring it into closer agreement with the details of Gödel's own formulation. Finally, we indicate what seems to be the main weakness of this emendation of Gödel's attempted proof.

Gaunilo observed against St. Anselm that his form of argument, if cogent, could be used to “prove” all sorts of unwelcome conclusions - for example, that there is somewhere a perfect island. It would seem to even follow that there are near-perfect, but defective, demi-gods and all matter of other theologically repugnant entities. Gaunilo concluded, reasonably enough, that something must be wrong with the argument.

Kurt Gödel's modern version of the Ontological Argument [12] involves an attempt to complete the details of Leibniz's proof that it is possible that there is a perfect being or a being with all and only “positive” attributes. Given this conclusion, other assumptions about positive properties, and, well, a second-order extension of the modal logic S5, Gödel successfully deduced the actual existence, indeed the necessary existence, of the being having all and only positive attributes. Alas, or “Oh, joy!”, depending on ones’ theological prejudices, J. Howard Sobel showed that Gödel's assumptions lead also to the conclusion that whatever is true is necessarily true. Followers of Spinoza aside, this casts quite considerable doubt on the premisses of the argument. We shall consider here Anderson's emendation which does not suffer from the mentioned defect and which is still recognizably closely related to Gödel's argument.

Here are the assumptions and definitions -the notion of a positive attribute is taken as a primitive by Gödel and in the present version. We hasten to add that the idea is not crystal clear; Gödel's own explanations are extremely terse and somewhat cryptic. A property's being positive is supposed to be a good thing, such properties being characteristic of a completely and necessarily non-defective being.