# Bernays on sets and classes

Prompted by his two pieces on Bernays as philosopher, I’ve found myself pausing before reading on in Parsons’s book to step sideways and remind myself about Bernays the set-theorist.

I confess I’d not before dipped into Bernays’s little 1958 book Axiomatic Set Theory (long available as a Dover reprint). This aims to give a first introduction to the kind of set theory Bernays had developed in that long series of papers ‘A System of Axiomatic Set Theory’ published in seven instalments in the JSL between 1937 and 1954 (a promised second volume elaborating on further issues from those papers never appeared).

What we get here is, of course, NBG set theory, with classes as well as sets — which is certainly familiar to those of us who learnt our logic from the first edition of Mendelson’s textbook. Now, Mendelson’s presentation of the motivation for this style of theory, as I recall it anyway, isn’t compelling. And certainly, careless classroom talk can make this style of theory seem puzzling (“if we can have a class of all sets, why not a superclass of all classes — isn’t it just ad hoc to allow one but not the other?”). So I was struck by Bernays’s lucid explanation here — albeit in slightly fractured English — of what is going on. In summary,

Th[e] distinction between sets and classes is not a mere artifice but has its interpretation by the distinction between a set as a collection, which is a mathematical thing, and a class as an extension of a predicate, which in comparison with the mathematical things has the character of an ideal object. This point of view suggests also to regard the realm of classes not as a fixed domain of individuals but as an open universe, and the rules we shall give for class formation need not to be regarded as limiting the possible formations but as fixing a minimum of admitted processes for class formation.

In our system we bring to appear this conception of an open universe of classes, in distinction from the fixed domain of sets, by shaping the formalism of classes in a constructive way, even to the extent of avoiding at all bound class variables, whereas with regard to sets we apply the usual predicate calculus. So in our system the existential axiomatic method is joined with a constructive formalism.

By avoiding bound class variables we have also the effect that the class formation $\{x \mid \mathfrak{A}(x)\}$ is automatically predicative, i.e. not including a reference by a quantifier to the realm of classes … Further the conception of classes as ideal objects in distinction from the sets as proper individuals comes to appear in our system by the failing of a primitive equality relation between classes. (pp. 56-57)

And so on, in the same vein. There is, of course, helpful modern elucidation about NBG and other theories with classes as well as sets out there in the literature, in e.g. the last two appendices of Michael Potter’s Set Theory and its Philosophy. But I do wish I’d known Bernays’s own presentation a long time ago!

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