The Axiom of Multiple Choice and Models for Constructive Set Theory
We propose an extension of Aczel’s constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf’s type theory (hence acceptable from a constructive and generalised-predicative standpoint). In addition, it is strong enough to prove the Set Compactness Theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.
There is a distinctive stance in the philosophy of mathematics which is usually called “generalised predicativity”. It is characterised by the fact that it does not accept non-constructive and impredicative arguments, but it does allow for the existence of a wide variety of inductively defined sets. Martin-Löf’s type theory [martinlof84] expresses this stance in its purest form. For the development of mathematics, however, this system has certain drawbacks: the type-theoretic formalism is involved and requires considerable time to get accustomed to, and the lack of extensionality leads to difficult conceptual problems. Aczel’s interpretation of his constructive set theory CZF in Martin-Löf’s type theory [aczel78] overcomes both problems: the language of set theory is known to any mathematician and CZF incorporates the axiom of extensionality. For this reason, CZF has become the standard reference for a set-theoretic system expressing the “generalised-predicative stance”.
It turns out, however, that CZF is not quite strong enough to formalise all the mathematics which one would like to be able to formalise in it: there are results, in particular in formal topology, which can be proved in type theory and are perfectly acceptable from a generalised-predicative perspective, but which go beyond CZF. There seem to be essentially two reasons for this: first of all, type theory incorporates the “type-theoretic axiom of choice” and secondly, Martin-Löf type theory usually includes W-types which allow one to prove the existence of more inductively defined sets than can be justified in CZF alone. To address this, we will suggest in this paper an extension of which includes a form of choice and W-types, so that in it one can develop formal topology, while at the same time having good model-theoretic properties.
Let us take the second point first. Already in 1986, Peter Aczel suggested what he called the Regular Extension Axiom (REA) to address this issue [aczel86]. The main application of (REA) is that it allows one to prove the “Set Compactness Theorem”, which is important in formal topology (see [aczel06, bergmoerdijk10c]), but not provable in CZF proper. Here we suggest to take the axiom (WS) instead: for every function the associated W-type is a set. (This is not the place to review the basics of W-types, something which we have already done on several occasions: see, for example, [bergmoerdijk10b].) One advantage of this axiom over (REA) is that it directly mirrors the type theory. In addition, (WS) is easy to formulate in the categorical framework of algebraic set theory, so that one may use this extensive machinery to establish its basic preservation properties (such as stability under exact completion, realizability and sheaves), whereas for (REA) such a formulation does not seem to be possible. It has been claimed, quite plausibly, that (REA) has similar stability properties, but we have never seen a proof of this claim.
As for the lack of choice in CZF, the axiom which would most directly mirror the type theory would be the “presentation axiom”, which says that the category of sets has enough projectives. The problem with this axiom, however, is that is not stable under taking sheaves. Precisely for this reason, Erik Palmgren together with the second author introduced in [moerdijkpalmgren02] an axiom called the Axiom of Multiple Choice (AMC), which is implied by the existence of enough projectives and is stable under sheaves. This axiom (which we will discuss towards the end of this paper) is a bit involved and it turns out that on almost all occasions where one would like to use this axiom a slightly weaker and simpler principle suffices. This weaker principle is:
For any set there is a set of surjections onto such that for any surjection onto there is an and a function such that .
It is this axiom which we will call (AMC) in this paper, whereas we will refer to the original formulation in [moerdijkpalmgren02] as “strong (AMC)”. (Independently from us, Thomas Streicher hit upon the same principle in [streicher05], where it was called TTCA; on the nLab, http://ncatlab.org, the principle is called WISC.)
To explain the name “Axiom of Multiple Choice”, we remark that for a surjection , a choice function is a section of . On the other hand, a multi-valued choice function is a function , defined on a cover of , for which . Our axiom provides a (necessarily nonempty) family of domains sufficient to find such a multi-valued choice function for any surjection .
So this is our proposal: extend the theory CZF with the combination of (WS) and (AMC). The resulting theory has the following properties:
It is validated by Aczel’s interpretation in Martin-Löf’s type theory (with one universe closed under W-types) and therefore acceptable from a generalised-predicative perspective.
The theory is strong enough to prove the Set Compactness Theorem and to develop that part of formal topology which relies on this result.
The theory is stable under the key constructions from algebraic set theory, such as exact completion, realizability and sheaves.
It is the purpose of this paper to prove these facts. As a result, CZF + (WS) + (AMC) will be the first (and so far only) theory for which the combination of these properties has been proved. And as a consequence of stability, the methods from [bergmoerdijk10c] are applicable to it and one can show:
The theory satisfies various derived rules, such as the derived Fan Rule and the derived Bar Induction Rule.
Moreover, we will show that the theory has a certain robustness about it. Indeed, assuming that the ground model for CZF satisfies (AMC) and (WS), it is impossible to use the standard model-theoretic techniques to prove independence of (AMC) and (WS) from CZF. To express this more formally, let us say that an axiom in the language of is reflected by sheaf extensions (for example), if for any CZF-model , the axiom holds in as soon as it holds in some sheaf extension of . Then as a fourth property of our theory CZF + (WS) + (AMC) we have
The theory is reflected by the model constructions of exact completion, realizability and sheaves.
It should be noted that establishing the first property for CZF + (WS) + (AMC) is quite easy, because stronger axioms are verified by the type-theoretic interpretation: (REA) can be interpreted (that was the main result of [aczel86]) and (REA) implies (WS) (see [aczelrathjen01, page 5–4]), while (AMC) is an obvious consequence of the presentation axiom which is validated by the type-theoretic interpretation (see [aczel82]). Therefore it remains to establish the other properties in the list.
The contents of this paper are therefore as follows. First, we will show in Section 2 that the Set Compactness Theorem follows from the combination of (WS) and (AMC). Then we will proceed to show that these axioms are stable under and reflected by exact completion (Section 3), realizability (Section 4) and sheaves (Section 5). Throughout these sections we assume familiarity with the framework for algebraic set theory developed in [bergmoerdijk07a, bergmoerdijk08, bergmoerdijk09, bergmoerdijk10b]. Finally, in Section 6 we will discuss the relation of our present version of (AMC) with the earlier and stronger formulation from [moerdijkpalmgren02, rathjen06b] and with Aczel’s Regular Extension Axiom.
2 The Set Compactness Theorem
The purpose of this section is to prove that, in CZF, the combination of (WS) and (AMC) implies the Set Compactness Theorem. To state this Set Compactness Theorem, we need to review the basics of the theory of inductive definitions in CZF, which will be our metatheory in this section.
If is a class, we will denote by the class of subsets of and if is a set, we will denote by the class of surjections onto .
Let be a set. An inductive definition on is a subset of . If is an inductive definition, then a subclass of is -closed, if
whenever is in .
Within CZF one can prove that for every subclass of there is a least -closed subclass of containing (see [aczelrathjen01]); it is denoted by . The Set Compactness Theorem is the combination of the following two statements:
is a set whenever is.
There is a set of subsets of such that for each class and each there is a set such that and .
As said, the Set Compactness Theorem is not provable in CZF proper, but we will show in this section that it becomes provable when we extend CZF with (WS) and (AMC).
To prove the result it will be convenient to introduce the notion of a collection square. In the definition we write for any function and each ,
as is customary in categorical logic.
A commuting square in the category of sets