5.1 Axiomatic

There is a full range of axiomatic constructive and intuitionistic set theories, extending from the proof-theoretically weak, e.g., Harvey Friedman’s B, all the way to IZF, intuitionistic Zermelo–Fraenkel set theory, which is equiconsistent with full classical ZF. Each of these theories is inspired, in large part, by the familiar idiom and axioms of ZF. The first formalized and specifically intuitionistic set theory may have been Heyting’s [Reference Heyting24]; more recent research was stimulated by Myhill’s ideas in [Reference Myhill39]. All of them endeavor to capture a conception of set already on view, namely, that a set is a coin with two foundational sides. As a class, it is on the one side the extension of a mathematical predicate (perhaps with parameters); on the other side, as an object of set theory, its internal structure is a register of its locus, within a universe of sets, determined by the combinatorics of the membership relation.

Friedman’s [Reference Friedman21] constructive set theory B was intended to formalize the constructive mathematics of Bishop within sets of finite rank over $\mathbb {N}$ . With its restricted forms of induction over the natural numbers, separation, replacement, and exponentiation (the latter standing in for the familiar powerset axiom), B is relatively weak proof-theoretically. Beeson [Reference Beeson5, Reference Beeson6] shows that B is conservative over Heyting Arithmetic, HA. It features a restricted form of AC—with $\mathbb {N}$ a base—as an axiom. However, without such an axiom, no general form of AC is derivable in the reduced B, since the latter is a subtheory of classical ZF.

The intuitionistic small set theory SST in [Reference McCarty, Shapiro and Rathjen38] has as its standard model the collection of pure cumulative sets of finite rank. It is definitionally equivalent to HA. In SST, AC is a theorem, but this is because the intended objects of the theory are all finite and decidable.

Aczel’s [Reference Aczel1] contains axioms for CZF, a constructive version of Zermelo–Fraenkel set theory, as well as an interpretation of CZF in Martin-Löf’s intuitionistic type theory (see Section 7). Although CZF is consistent with the statement that $\mathbb {N}$ is a base, it does not derive it.

We shall need the following special case of AC, called $\mathbf {AC}_{\mathbb {N}, \mathbb {N}}$ :

$$\begin{align*}\forall x \in \mathbb{N} \ \exists y\in \mathbb{N} \ R(x,y) \ \rightarrow \ \exists f \in (\mathbb{N}\Rightarrow \mathbb{N})\ \forall x\in\mathbb{N} \ R(x,f(x)).\end{align*}$$

The strongest of the set theories listed above is IZF. It is relatively consistent with the invalidity of LEM [Reference Grayson23], with Brouwer’s Theorem [Reference Fourman and Hyland20], and with intuitionistic Church’s Thesis [Reference McCarty37]. As a subtheory of classical ZF, IZF fails to derive AC, and even $\mathbf {AC}_{\mathbb {N}, \mathbb {N}}$ . In IZF, as well as in its subtheories CZF and B, it is consistent that AC $_{\mathbb {N}, \mathbb {N}}$ fails. The Goodman–Myhill proof can be carried out in IZF, but not in its full generality in CZF and B, which feature restricted, rather than full, separation. Since both of these contain pairing as an axiom, however, the proof that $\{ a, b\}$ is a base just in case identity on it is decidable can be carried out in either system.

5.2 Brouwer’s original

Brouwer published repeatedly on an intuitionistic set theory, e.g., [Reference Brouwer12], but never offered, in his published works at least, a full axiomatic treatment of sets. He did, however, describe at least two concepts of sets or set-like collections: species in general and species arising from spreads. Neither of these demonstrates, or even suggests, that AC should be true.

First, a Brouwerian species is the extension of an extensional mathematical predicate, provided that the predicate does not employ any impredicative specification. Species are to be individuated extensionally, in the sense that species with the same members are the same:

Moreover, elementhood in a species is to be extensional, in the sense that if a belongs to species S and $a = b$ , then b belongs to S, too:

Brouwer explicitly allowed there to be species of higher type—species whose members are other species.

Brouwerian spreads are trees of sequences. In particular, a spread T is an inhabited, decidable collection of finite sequences of natural numbers satisfying two conditions:

1. 1. It is closed under predecessor: if a sequence s belongs to T, then so does every initial subsequence t of s.

2. 2. Each sequence s in T has at least one immediate successor: if $s \in T$ , then there is at least one sequence $t\in T$ extending s by a single component.

The species of members of a spread T consists of all the sequences each of whose initial segments belongs to T.Footnote ⁴

Both these conceptions—species and spread—can be elaborated and developed theoretically ad infinitum, without lending any support either to AC in general or even to AC $_{\mathbb {N}, \mathbb {N}}$ . For instance, an intuitionistic theory of species satisfying full comprehension can be constructed over any model of the intuitionistic set theory IZF and then proven to be independent of AC for any sets and classes in the conjoint universe. Were AC assumed to hold for classes, it would also hold for sets and, by the Goodman–Myhill proof, would imply LEM.

5.3 Bishop

Bishop adopted a distinctively constructivist concept of set, writing,

Bishop elaborated on the conception—with its double specification, one for membership and one for equality—several pages later:

Bridges and Vîţă, distinguished followers of Bishop, explain why, on this “double specification” notion of set, one should not expect AC to hold generally over sets.Footnote ⁵ They use the subscripted sign $=_{X}$ to denote an equality allied with a set X.

[T]he hypothesis

$$\begin{align*}\forall x\in X\ \exists y \in Y\ (\langle x,y\rangle \in S)\end{align*}$$

of AC means that we have an algorithm that, applied to each element x of X and the data showing that x belongs to X, constructs an element y of Y and demonstrates that $\langle x,y\rangle \in S$ . This much is clear. However, there is no guarantee that the algorithm will respect the equality relation on X—in other words, that if $x =_{X} x'$ , and the algorithm constructs y and $y'$ in Y such that $\langle x, y\rangle \in S$ and $\langle x',y'\rangle \in S$ , then $x' =_{Y} y'$ . Indeed, we should expect that the computation of y might use data that are associated with properties intrinsic to x that do not apply intrinsically to $x'$ . [Reference Bridges and Vîţă11, p. 17]

Bridges and Vîţă do allow that $\mathbb {N}$ is a base, but, in their mathematical work, that assumption features as a separate postulation and not as a direct or indirect consequence of Bishop’s conception of set itself. The assumption that $\mathbb {N}$ is a base is consistent with the background set theory, since it holds in the Kleene realizability universe for IZF.

Once again, the foregoing argument against AC depends upon a plain extensionality requirement: that functions on sets preserve identity.

5.4 Kleene–Kreisel–Troelstra

The concept of realizability, crucial to the current metatheoretic understanding of intuitionism at every level, originated in a brilliant paper by Kleene [Reference Kleene28]. Kleene’s interpretation of the formulas of intuitionistic arithmetic associates those formulas with collections of natural numbers generally treated as codes or indices of Turing machines. The association is specified recursively by laying down conditions, retracing the structures of formulas, under which a natural number n realizes formula $\phi $ or, in symbols, $n \Vdash \phi $ . For instance, for the material implication, $n \Vdash (\phi \rightarrow \psi )$ just in case

$$\begin{align*}\forall m \ (\text{if }m \Vdash \phi, \text{ then } \{n\}(m) \text{ is defined and } \{n\}(m)\Vdash \psi),\end{align*}$$

that is, for any m that realizes the antecedent $\phi $ , the Turing machine n computing the function $\lambda x. \{ n\}(x)$ applies to m and outputs a realizer (or a realizability witness) for the consequent $\psi $ .

A premier way to extend Kleene’s original idea to set theory and higher-order systems is by working with a two-part concept of set introduced by Kreisel and Troelstra [Reference Kreisel and Troelstra29]. First, realizability sets are defined: a realizability set S is any set of pairs $\langle n, x\rangle $ wherein the n’s are natural numbers and the x’s are members of some set A. In other words, a realizability set is a subset of the Cartesian product $\mathbb {N} \times A$ . When $\langle n, x\rangle $ belongs to the realizability set S, we write

$$\begin{align*}n \Vdash x \in S.\end{align*}$$

Second, when it comes to quantification over all realizability sets, we say that

$$\begin{align*}n \Vdash \forall s \ \phi(s) \text{ if and only if } \forall s \ n\Vdash \phi(s).\end{align*}$$

In short, quantification over realizability sets exerts no effect—or, in the jargon, is “uniform”—over the realizability relation $\Vdash $ .

A formula $\phi $ of the set-theoretic language holds under realizability just in case $\exists n \ n \Vdash \phi $ . The entire set theory IZF is sound with respect to this concept of realizability for sets: provably within IZF, for all sentences $\phi $ in the language of set theory, if IZF $\ \vdash \phi $ , then $\exists n \ n\Vdash \phi $ . The law of excluded middle fails outright under the realizability interpretation:

$$\begin{align*}\exists n \ n \Vdash \neg \forall p \ (p \vee \neg p).\end{align*}$$

Consequently, by Goodman–Myhill, AC is false in the realizability interpretation of IZF.

7.1 Intensional Axiom of Choice

Howard [Reference Howard27] indicated how the correspondence between propositions and types that would later be called the Curry–Howard correspondence yields a choice function from an assumed proof of $\forall x\in \mathbb {N} \ \exists y\in \mathbb {N} \ R(x,y)$ . The formal language studied by Howard was, however, not rich enough to provide for the definition of such a function. Only with the more thoroughgoing implementation of the Curry–Howard correspondence by Martin-Löf in [Reference Martin-Löf34] did a choice function become formally definable, and then not only over $\mathbb {N}$ , but over any quantificational domain of Martin-Löf’s type theory.

The Curry–Howard correspondence correlates notions of logic with notions of type theory. In particular, propositions are identified as types: proposition A is identified as the type of proofs of A. By a proof of a proposition A one then understands an object making A true and not an act by means of which one convinces oneself of the truth of A. It is natural to think of a proof in this sense as a truthmaker of A, as noted by Sundholm [Reference Sundholm45]. Indeed, that a proposition is true means, under the Curry–Howard correspondence, that it is inhabited as a type, that a proof of it exists.

Just as one can define a type by saying what it is to be an object of that type, so one can define a proposition by saying what it is to be a proof of that proposition. Explaining propositions in this way is, in effect, just what the BHK explanation does; hence one sees the close relationship between it and the Curry–Howard correspondence. Curry–Howard, however, adds precision by laying down how a (canonical) proof of a proposition is formed from its parts. In particular, a proof of an existential proposition, $\exists x\in A \ C(x)$ , is stipulated to be a pair $\langle a,p\rangle $ where a is an element of the domain of quantification, A, and p is a proof of $C(a)$ . A proof of a universal proposition, $\forall x\in A \ C(x)$ , is stipulated to be a lambda term, $\lambda x.p(x)$ , where $p(a)$ is a proof of $C(a)$ for every a in the domain of quantification, A.

When defining a type A in intuitionistic type theory, it is in fact not enough just to say what it is to be an object of type A. In addition, one must say what it is to be identical objects of type A. In a terminology sometimes used by philosophers: to define a type one must give both a criterion of application and a criterion of identity (cf. Bishop’s notion of set, Section 5.3). Every type therefore comes equipped at birth with a notion of identity, which is usually expressed in the form $a=b : A$ (this could be read as “a is the same A as b”). It then makes sense to speak of functions on types: a function, $f(x)$ , from A to B is a transformation such that $f(a)=f(b): B$ whenever $a=b : A$ . All open terms in intuitionistic type theory are functions in this sense. It follows that if $p(a)$ is a proof of $C(a)$ for every a in the domain of quantification, A, then $p(a)=p(b):C(a)$ whenever $a=b : A$ .

Under the Curry–Howard correspondence, the formation of $\lambda x.p(x)$ as a proof of $\forall x\in A \ C(x)$ corresponds to an application of the $\forall $ -introduction rule in natural deduction. Corresponding to an application of the $\forall $ -elimination rule is the formation of a term of the form $\mathbf {ap}(c,a)$ , whenever c is a proof of $\forall x\in A \ C(x)$ and a is an element of A. Whereas the $\lambda $ -operator is primitive, the $\mathbf {ap}$ -operator is defined, namely by the equation

$$\begin{align*}\mathbf{ap}(\lambda x.p(x), a) = p(a) : C(a).\end{align*}$$

Under the Curry–Howard correspondence, this equation corresponds to the $\forall $ -reduction rule of natural deduction introduced by Prawitz [Reference Prawitz41].

The BHK clauses for implication, $A\rightarrow B$ , and the universal quantifier, $\forall x\in A \ C(x)$ , both invoke the idea of a transformation, in the former case from A to B, in the latter case from any a in A to $C(a)$ . In intuitionistic type theory, implication comes out as a special case of universal quantification, namely the case where each $C(a)$ is one and the same proposition, B. In particular, the $\lambda $ -operator can be used to construct a proof, $\lambda x.p(x)$ , of $A\rightarrow B$ provided $p(a)$ is a proof of B whenever a is a proof of A.

In light of the $\mathbf {ap}$ -operator and the functionality of open terms, we may think of $\lambda x.p(x)$ as a function from A into the family of types (or propositions) $\{C(x)\}_{x\in A}$ . At the end of the previous section we noted that for the BHK explanation to justify AC, the construction that serves as a proof of a universally quantified proposition must be a function. This requirement is now met. To obtain a choice function from a proof of the antecedent, $\forall x\in A \ \exists y \in B \ R(x,y)$ , of AC, however, also projection functions are required.

Just as $\forall $ -elimination corresponds to the $\mathbf {ap}$ -operator, so $\exists $ -elimination corresponds to a certain operator. We shall not describe the precise workings of this operator here—suffice it to say that from it, one can define first and second projection functions,

$$\begin{align*}\mathbf{fst}(\langle a,b\rangle) = a\quad\quad\quad \mathbf{snd}(\langle a,b\rangle) = b.\end{align*}$$

Given a proof c of an existential proposition, $\exists x\in A \ C(x)$ , $\mathbf {fst}(c)$ is an element of the domain of quantification, A, and $\mathbf {snd}(c)$ is a proof of the proposition $C(\mathbf {fst}(c))$ . It was in specifying the type of $\mathbf {snd}(c)$ that Howard hit the limits of his formal system. One needs for that purpose to have access to the notion of a propositional function over the proofs of a proposition. In particular, one needs access to the propositional function $C(\mathbf {fst}(z))$ over proofs z of $\exists x\in A \ C(x)$ . This clearly requires a thoroughgoing implementation of the Curry–Howard correspondence.Footnote ⁷

Suppose c is a proof of an instance of the antecedent of AC, that is, of a proposition $\forall x\in A \ \exists y\in B \ R(x,y)$ , and let a be an element of the domain A. Then $\mathbf {ap}(c,a)$ is a proof of $\exists y\in B \ R(a,y)$ . Using the two projection functions, we find

$$\begin{align*}\mathbf{snd}(\mathbf{ap}(c,a)) \in R(a, \mathbf{fst}(\mathbf{ap}(c,a))).\end{align*}$$

By means of the $\lambda $ -operator, we can then form a choice function

$$\begin{align*}\lambda x.\mathbf{fst}(\mathbf{ap}(c,x)) \in (A\Rightarrow B).\end{align*}$$

Given these ingredients, it is now an easy exercise to construct the full proof of AC (for details, see [Reference Martin-Löf35]). The proof goes through also if the co-domain of the choice function is a family of types, $B(x)$ , depending on the domain, A, or more precisely, if the choice function belongs to the generalized Cartesian product $\prod _{x\in A} B(x)$ . This more general form of the Axiom of Choice includes, as a special case, the formulation that the Cartesian product of a family of non-empty types is non-empty.

Any type that may serve as the domain of the universal quantifier in intuitionistic type theory is therefore a base. Included among these are $\mathbb {N}$ as well as any type that can be formed from $\mathbb {N}$ by means of product, disjoint union and exponentiation, such as $\mathbb {N}\times \mathbb {N}$ , $\mathbb {N}\Rightarrow \mathbb {N}$ , and $(\mathbb {N}\Rightarrow \mathbb {N})\Rightarrow \mathbb {N}$ . The domain $\mathbb {R}$ of real numbers, however, remains out of reach. A domain Cauchy of Cauchy sequences of fractions can be defined as a type (the canonical elements of this type will be pairs $\langle s,p\rangle $ , where s is a sequence of fractions and p is a proof that s is Cauchy). The construction needed to obtain $\mathbb {R}$ from Cauchy, however, is not among the type-forming operations of (standard) intuitionistic type theory.

We obtain the domain $\mathbb {R}$ from Cauchy by, in effect, identifying certain Cauchy sequences that are not identical according to the identity relation native to the type Cauchy. A standard way of implementing this construction is by the formation of a quotient structure. Another way is to let co-convergence simulate identity on Cauchy. The type Cauchy will continue to have its native identity relation, but we require that all functions defined on Cauchy must respect co-convergence, so that co-convergence in effect plays the role of identity on Cauchy. Martin-Löf follows the latter course in clarifying why there is no contradiction between the provability of AC in his system and results such as Diaconescu’s Theorem.Footnote ⁸

7.2 Extensional Axiom of Choice

It may be difficult to pin down in the abstract just what extensionality amounts to, beyond a certain family resemblance among certain principles. In intuitionistic type theory it is natural to say that an extensionality principle is a principle that identifies elements of a type A that are not identical according to the identity relation native to A. Likewise, an extensional construct is a construct that renders elements of A identical that are not identical according to the identity relation native to A.

Function extensionality, according to which $\lambda x.p(x)$ and $\lambda x.q(x)$ are identical elements of $A\Rightarrow B$ if and only if $p(a)$ and $q(a)$ are identical for every $a\in A$ , is one example of an extensionality principle. Another example is the principle according to which propositions are identical if and only if they are materially equivalent. The type $\mathbb {R}$ , regarded as the type Cauchy equipped with co-convergence as the identity relation is an extensional construct, and so is Aczel’s constructive version of the cumulative hierarchy of sets serving as a model of CZF [Reference Aczel1–Reference Aczel3]. The model consists of a type V together with a relation $\sim _{\mathbf {V}}$ of extensional equality. From the definition of V, it is clear that it contains infinitely many empty sets. The relation $\sim _{\mathbf {V}}$ identifies all of these and, more generally, any two sets a and b such that $\forall x \in a \ \exists y \in b \,(x\sim _{\mathbf {V}} y)$ and $\forall y \in b \ \exists x\in a \,(x\sim _{\mathbf {V}} y)$ .

In light of this understanding of extensionality, we see that it was natural for Martin-Löf to call an extensional set a type A equipped with an equivalence relation that is to simulate identity on A.Footnote ⁹ Where $\sim $ is the equivalence relation in question, the extensional set may be written as the pair $(A,\sim )$ . Sets in the sense of set theory are extensional, not only in the sense that the set-theoretical Axiom of Extensionality holds for them, but also in the sense that extensional constructs, such as quotient formation and naive separation,Footnote ¹⁰ preserve sethood.

With the notion of extensional set in hand, we can formulate the extensional axiom of choice, ExtAC, which requires that the choice function respects the equivalence relations $\sim _A$ and $\sim _B$ simulating identity on A and B:

$$\begin{align*}\forall x\in A \ \exists y\in B \ R(x,y) \ \rightarrow \ \exists f : \big((A,\sim_A) \Rightarrow (B,\sim_B)\big) \ \forall x\in A \ R(x,f(x)).\end{align*}$$

By the notation $f : \big ((A,\sim _A)\Rightarrow (B,\sim _B)\big )$ we mean that f is a function from A to B that respects the equivalence relations $\sim _A$ and $\sim _B$ in the sense that $\forall x,y \in A \ (x\sim _A y\rightarrow f(x)\sim _B f(y))$ is true. Since we are in the setting of extensional sets, it is natural to require that R be an extensional relation, that is, that $R(x,y)$ and $R(x',y')$ be materially equivalent whenever $x\sim _A x'$ and $y\sim _B y'$ hold.

As Martin-Löf [Reference Martin-Löf36] makes clear, it is ExtAC that corresponds in intuitionistic type theory to the Axiom of Choice in set theory. Apart from the conceptual argument that a set in the sense of set theory is better understood as an extensional set $(A, \sim )$ than as a type A, there is also a compelling technical argument: when ExtAC is assumed, Aczel’s model V validates (set-theoretical) AC.

The constructive unacceptability of ExtAC is borne out by Carlström [Reference Carlström15], who shows that, in intuitionistic type theory, it entails the Law of Excluded Middle.Footnote ¹¹ Indeed, we have no reason to expect that the choice function constructed from a proof of $\forall x\in A \ \exists y\in B \ R(x,y)$ is extensional in the sense required by ExtAC. That extensionality is the culprit here is nicely illustrated by the fact that ExtAC is just one of many extensionality principles which when added to intuitionistic type theory yields classical logic [Reference Lacas and Werner30, Reference Maietti32, Reference Valentini52].

Although ExtAC is constructively unacceptable, it is a theorem of intuitionistic type theory that a choice operation always exists: this is just what intensional AC becomes in the setting of extensional sets. An operation, O, between extensional sets $(A,\sim _A)$ and $(B,\sim _B)$ is here simply a function from A to B: it respects the identity relations native to A and B, but need not respect the equivalence relations $\sim _A$ and $\sim _B$ .

7.3 The Presentation Axiom

On each type, A, there is a binary propositional function of identity, $=_A$ , defined as the smallest reflexive relation on A. In the setting of extensional sets, where an arbitrary equivalence relation on A may serve as identity, we may call the relation $=_A$ the true identity relation on A. Let us call an extensional set of the form $(A,=_A)$ an intensional-extensional set. All intensional–extensional sets are bases. Moreover, any extensional set $(A,\sim )$ is the surjective image of an intensional–extensional set, namely the image of $(A,=_A)$ under the identity function on A. (If $(A,\sim )$ is regarded as a quotient structure, then the map in question is the natural one.) Extensional sets known to be bases—the intensional–extensional sets—thus form a subdomain of all extensional sets, and every extensional set is the surjective image of an extensional set from this subdomain.

Abstracting this situation from the type-theoretical framework, we arrive at the so-called Presentation Axiom, PAx: every set is the surjective image of a base.Footnote ¹² The axiom holds in Aczel’s V, and hence it is consistent with CZF and, moreover, constructively acceptable. It is not, however, a theorem of CZF. Indeed, it is not even a theorem of classical ZF plus the axiom of dependent choices [Reference Rathjen42]. Aczel’s proof [Reference Aczel2] shows, more concretely, that, in V, (i) sets formed by means of set-theoretical correlates of the standard type-forming operations of intuitionistic type theory are all bases and (ii) every set is the surjective image of such a set. The proof thus gives expression to the idea that the quantificational domains of type theory—all of which are bases—form a subdomain of all sets, some of which fail to be bases.