Universes in Toposes

We discuss a notion of universe in toposes which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families of types in such a universe by structural recursion and to quantify over such families. Further, we show that (hierarchies of) such universes do exist in all sheaf and realizability toposes but neither in the free topos nor in the Vω+ω model of Zermelo set theory. Though universes in Set are necessarily of strongly inaccessible cardinality it remains an open question whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF). The background information about toposes and fibred categories as needed for our discussion in this paper can be found e.g. in the fairly accessible sources [MM, Jac, Str2]. 1 Background and Motivation It is commonly agreed on that elementary toposes with a natural numbers object (NNO) provide a concise and flexible notion of model for constructive Higher Order Arithmetic (HAH). Certainly, a lot of mathematics can be expressed within HAH. So what is the need then for set theory (ZFC) which is generally accepted as the foundation for mainstream mathematics? Well, ZFC is much stronger than HAH in the following respects: (1) ZFC is based on classical logic whereas HAH is based on the weaker intuitionistic logic. (2) ZFC postulates the axiom of choice whereas HAH does not. (3) ZFC postulates the axiom of replacement which cannot even be formulated in HAH. 1Notice, however, that the axiom of replacement obtains its full power only in presence of The logic of toposes (with NNO) is inherently intuitionistic and in HAH the axiom of choice implies classical logic. Therefore, we have to give up (1) and (2) above when considering Grothendieck and realizability toposes as models of some kind of set theory. But what about (3), the axiom of replacement? First of all notice that there are models of set theory without replacement but satisfying classical logic and choice, namely Vω+ω. 2 On the other hand a lot of toposes, in particular Grothendieck toposes and realizability toposes, do model the axiom of replacement whereas in most cases they refute classical logic and the axiom of choice. More precisely, the above mentioned toposes model IZF, i.e. ZF with intuitionistic logic and axiom of regularity reformulated as ∈-induction. Though a large class of toposes validates IZF one still may complain that the formulation of IZF suffers from “epsilonitis”, i.e. that it “implements” informal mathematics via the ∈-relation rather than axiomatizing mathematical practice in terms of its basic notions. So one may ask what is the mathematical relevance of the set-theoretic replacement axiom? Maybe a set-theorist would answer “for constructing ordinals greater than ω+ω” which, however, may seem a bit disappointing because most mathematics can be formulated without reference to transfinite ordinals. Actually, what axiom of replacement is mainly needed for in mathematical practice is to define families of sets indexed by some set I carrying some inductive structure as, typically, the set N of natural numbers. For example, most mathematicians would not hesitate to construct the sequence (P(N))n∈N by (primitive) recursion over N. Already in ZC, however, this is impossible because {〈n,P(N)〉 | n∈N} 6∈ Vω+ω. Usually, in ZFC the sequence (P(N))n∈N is constructed by applying the axiom of replacement to an appropriately defined class function from the set of natural numbers to the class of all sets. However, in a sense that does not properly reflect the mathematician’s intuition who thinks of (P(N))n∈N as a function f from N to sets defined recursively as f(0) = N and f(n+1) = P(f(n)). There is, however, a “little” problem, namely that the collection of all sets does not form a set but a proper class. Notice, however, that a posteriori the image of f does form a set the full separation scheme. In recent, yet unpublished work by S. Awodey, C. Butz, A. Simpson and T. Streicher [ABSS] it has been shown that set theory with bounded separation, i.e. separation restricted to bounded formulas, but with replacement (and even strong collection) is equiconsistent to HAH as long as the underlying logic is intuitionistic. Otherwise the classical Principle of Excluded Middle allows one to derive full separation from replacement. In the context of this paper when we say replacement we mean the power of replacement together with full separation (although the latter does not make sense from a type-theoretic point of view!). 2But notice that Vω+ω validates ZC, i.e. ZFC without replacement, which, however, is still stronger than HAH as already Z proves the consistency of HAH. 3Axiom of regularity and ∈-induction are equivalent only classically as in IZF the principle of excluded middle follows from the axiom of regularity just as in HAH the principle of excluded middle follows from the least number principle. 4There are notable exceptions typically in the area of descriptive set theory as e.g. Borel determinacy which is provable in ZF (as shown by D. A. Martin) but not in Z (as shown by H. Friedman). Even IZF does not decide Borel determinacy as it holds in Set but not in Hyland’s effective topos Eff .