nLab Scott's trick

Idea

Scott’s trick (due to Dana Scott) is a technical device in a material set theory such as ZF, that is used to construe a quotient of a proper class as another class.

Discussion

We work throughout with the theory ZF, leaving it to the reader to consider how the discussion applies to other material set theories of comparable strength.

A “class” is not a formal object of ZF, but a meta-object defined by a formula of ZF such as $x \notin x$. (Formally, then, a class is a formula considered up to logical equivalence.) To some extent such meta-objects can be handled like “sets” (the formal objects), so long as constructions remain safely within the confines of first-order, not higher-order logic. For example, there are various ZF hacks for defining the product of two classes as a class, the sum of two classes as a class, etc., but there is no way of forming exponentials of classes, or the power object of a class. In technical terms, the meta-category of classes and class functions may be regarded as a pretopos, but not a topos.

A critical case here is forming the quotient of a class by an equivalence relation. In the usual development of axiomatic set theory, the quotient of a set $X$ by an equivalence relation $\sim$ on $X$ is located within the power set $P X$: each $\sim$-equivalence class is an element of $P X$, and the set of equivalence classes is thus rendered as a definable subset of $P X$. This naive approach to defining quotients is not available for classes $C$, since in the first place we do not have a power class $P C$ to work with.

As a workaround, Dana Scott introduced the following trick. In ZF set theory, we define the ordinal rank of a set by transfinite induction (recursion), giving a class function $V \to Ord$. This uses the axioms of foundation and of replacement. The collection of sets of ordinal rank $\alpha$ or below is itself a set $V_\alpha$; this is the underpinning of the cumulative hierarchy picture of set theory.

Thus, given an equivalence relation $\sim \subseteq C \times C$ on a class $C$, we may consider in each equivalence class just those sets whose rank is the minimum $\alpha$ within that class. The collection of those sets is again a set, a subset $c$ of $V_\alpha$, by the axiom of separation. We let those sets $c$ proxy as equivalence classes; those sets form a class, and this class is the desired quotient $Q$. Indeed, there is a quotient map $C \to Q$, taking $x \in C$ to the set of smallest-rank sets $\sim$-equivalent to $x$, and one may argue that this quotient map has the requisite universal property.

Applications

There are many applications of Scott’s trick. See for instance

• localization

where the general construction for a large category involves passage to a quotient, and also

• ultraproduct

where standard discussions of ultrapowers of class-sized models take this technical trick into account.

References

The eponymous trick was introduced in a conference:

• Dana Scott, Definitions by abstraction in axiomatic set theory, Meeting of the American Mathematical Society, University of British Columbia, Vancouver, Canada (June 1955). (web)

Scott’s immediate application was to give a viable notion of cardinal number, not as a von Neumann ordinal which requires the well-ordering principle, but via a proxy of the equivalence class for the equivalence relation “is in bijection with”, as explained above.

A brief textbook account (not mentioning Scott’s name) is in

• Thomas Jech, Set Theory, 3rd millennium (revised) ed., Springer Monographs in Mathematics, Springer (2003).

See particularly page 65.