nLab predicative mathematics

Predicative mathematics

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Mathematics

Predicative mathematics

Idea

Predicative mathematics is a way of doing mathematics without allowing impredicative definitions.

Informally, a definition is impredicative if it refers to a totality which includes the thing being defined. For example, the definition of a particular real number xx as the least upper bound of a given set AA is impredicative, because it characterizes xx as a particular element of some set (the upper bounds of AA) which includes xx. Possibly xx can be defined in some other way and then proved to be the least upper bound of AA, but to define it thus by fiat is impredicative.

There are (at least) two broad schools of the foundations of predicative mathematics that don't talk much to each other: one school that uses lower-order forms of higher-order logic (or the set theory that this justifies) and classical logic, and a constructive school that uses first-order type theory (or the set theory that this justifies) and intuitionistic logic. The common ground is that both schools reject power sets; other axioms may vary. Here we tend to think of what predicative mathematics allows the category of sets to be.

Constructive predicativists sometimes accept principles (such as function sets and the axiom of replacement) that classical predicativists must reject because they imply impredicative results using excluded middle. Such mathematics may be called weakly predicative.

Impredicative axioms

Not all of these axioms are rejected by all predicativists, but they at least come under some suspicion.

Infinity

The axiom of infinity is not usually considered impredicative, but we list it anyway, as it is needed for the others to have force. Mathematics that does not require this axiom, finite mathematics, can be interpreted in a predicative framework even if it uses many of the axioms below.

Power set

The axiom that any set has a power set is perhaps the fundamental feature missing from predicative mathematics. In particular, the sequence

N,𝒫N,𝒫𝒫N, \mathbf{N}, \mathcal{P}\mathbf{N}, \mathcal{P}\mathcal{P}\mathbf{N}, \ldots

may be accepted in part, but not forever.

The failure of the power set axiom means that the category of sets is not an elementary topos.

Unbounded separation

The constructive school generally accepts the axiom of replacement but not unbounded forms of the axiom of separation. (This choice is not available to the classical school, since replacement and excluded middle together imply full separation.)

Naïvely, the axiom of separation says that, if AA is a set and PP is a function from AA to the set of truth values, then there is a set

{A|P}={xA|P(x)}. \{ A | P \} = \{ x \in A \;|\; P(x) \} .

To be precise, however, this PP should be written as a predicate in the language of set theory. The form of separation justified by type theory and such structural set theories as ETCS requires the quantifiers in this predicate to be guarded by sets; unbounded separation is the generalisation of this to arbitrary quantifiers.

Arguably, the impredicative core of both separation and power sets (in the presence of bounded separation) is limited separation: separation in which the quantifiers in PP may be guarded by power classes.

We need more on this, particularly with regards to the classical school and replacement.

Function sets

One sometimes speaks of forbidding function sets instead of power sets. That is, it is the sequence

N,N N,N N N, \mathbf{N}, \mathbf{N}^{\mathbf{N}}, \mathbf{N}^{\mathbf{N}^{\mathbf{N}}}, \ldots

that is avoided.

Of course, function sets can be constructed out of power sets (using bounded separation), so forbidding function sets certainly forbids power sets. The converse holds if there is a set Ω\Omega of truth values.

With excluded middle, the set of truth values is easy to achieve, as {0,1}\{0,1\}; in particular, if you have N\mathbf{N}, then you certainly have Ω\Omega. So the classical school of predicativism rejects function sets.

The constructive school, however, often accepts function sets (thus being weakly predicative). In this school, the sequence above is fine. Actually, the slightly stronger axiom of subset collection is adopted by Peter Aczel's CZFCZF and justified by Per Martin-Löf's ITTITT.

Brouwer, on the other hand, did not accept the sequence above, although his followers differ on when (if ever) it stops.

Excluded middle

Classical predicativists of course accept excluded middle; otherwise they would be constructivists. But from the perspective of weakly predicative constructive mathematics, excluded middle is impredicative, since it implies power sets (given function sets) and unbounded separation (given replacement).

Propositional resizing

In the constructive school, one would sometimes have multiple sets of propositions, but those only represent the set of UU-small propositions or subsingletons Ω U\Omega_U, given a universe UU, rather than the set of all propositions Ω\Omega. In general, given any two universes UU and VV, one cannot prove that Ω U\Omega_U is equivalent to Ω V\Omega_V. This is common in type theoretic models of constructive mathematics.

The axiom of propositional resizing is then statement that given any two universes UU and VV, the sets Ω U\Omega_U and Ω V\Omega_V of propositions in UU and VV respectively are in bijection with each other Ω UΩ V\Omega_U \cong \Omega_V. This axiom implies that there is only one set of propositions up to bijection, which in combination with the existence of function sets imply power sets, so is impredicative.

Ill-founded structures

Most foundations of mathematics are predicative in one sense: no set may belong to itself. This (or rather, a certain strengthening of this) is the axiom of foundation. An alternative is the axiom of antifoundation, which explicitly allows for and tames such sets as \bullet, where ={}\bullet = \{\bullet\}. Indeed, this equation is a perfectly good way to define \bullet using antifoundation, yet this is about as impredicative as a definition can get.

Once one accepts the axiom of infinity, there's not much objection to accepting more general inductive types such as inductive-inductive types and quotient inductive types; these are sets that are defined recursively much like a natural numbers object. Categorially, we may see these as initial algebras of certain functors on SetSet. Coinductive types, which are the final coalgebras of these functors, also exist in impredicative theories, but not predicatively.

Polymorphic definitions

A different sort of impredicativity is to be found in some type theories, in which, roughly speaking, one is allowed to define “functions” whose “domain” is the class of all types. For instance, one might be able to form a type called α:Type,αα\forall \alpha:Type, \alpha\to\alpha, which has the property that an inhabitant of that type is a “function” which assigns to every type α\alpha, an endomorphism of α\alpha. This is clearly impredicative, since the type α:Type,αα\forall \alpha:Type, \alpha\to\alpha is also a possible value of α\alpha.

The philosophy behind this sort of impredicative definition is that any inhabitant of such a type must be defined “uniformly” enough that it uses no details about the type α\alpha, and thus can equally well be applied to any α\alpha. For instance, consider the operation which assigns to every α\alpha the identity id:ααid\colon \alpha\to\alpha; this is defined in exactly the same way for every α\alpha, and hence inhabits the type α:Type,αα\forall \alpha:Type, \alpha\to\alpha.

This sort of impredicativism can be shown to be incompatible with impredicative set-theoretic axioms such as power sets; see this paper of Andy Pitts. Since such type theories generally do have function types, it follows that they cannot be classical.

The category of sets

So what is the category of sets in predicative mathematics?

At bottom, let us suppose that SetSet is a Heyting pretopos; this is a category whose internal logic is first-order and contains only constructions that don't require any of the above axioms.

Since we're not doing finite mathematics, we may also include a natural numbers object. In fact, we could include more general inductive types, since these are no harder to justify philosophically than N\mathbf{N}, although the proofs that these exist if N\mathbf{N} does rely on possibly impredicative axioms. Then SetSet is a Heyting WW-pretopos.

If you accept function sets, then SetSet is locally cartesian closed and thus a Π\Pi-pretopos. In this case, N\mathbf{N} is enough to get all WW-types, so we have a Π\Pi-WW-pretopos. If you accept excluded middle, then SetSet is a Boolean pretopos or even a Boolean WW-pretopos. But a Boolean Π\Pi-pretopos is necessarily a topos, which would make the theory impredicative.

However, SetSet is still a Grothendieck topos, defined as a category of sheaves or in terms of Giraud's characterisation. We require the existence of power sets to prove the theorem that such a category is an elementary topos, so predicatively a Grothendieck topos may not be an elementary topos at all.

The real numbers

An important question in predicative mathematics is the status of the set R\mathbf{R} of real numbers. This set is often constructed as a subset R DR_D of 𝒫N\mathcal{P}\mathbf{N} or as a subquotient R CR_C of N N\mathbf{N}^{\mathbf{N}}, neither of which can be formed in an arbitrary Heyting WW-pretopos. The latter can be formed in a Π\Pi-WW-pretopos, but it is not necessarily correct.

The constructive school of predicativism can construct R\mathbf{R} in various ways. One method is to use R CR_C directly, but this will only go so far unless something is done to prove that it is Dedekind-complete. This will follow from weak countable choice (WCCWCC), which is accepted by most constructive schools. Using subset collection, a variation on R CR_C is possible which can be proved Dedekind-complete without WCCWCC.

However, not all constructive predicative mathematics accept subset collection or weak countable choice. Another method is to use a predicative version of R DR_D, where given a σ \sigma -frame Σ\Sigma, Σ\Sigma-Dedekind cuts are defined as pairs of functions L,UL, U from the rational numbers \mathbb{Q} to Σ\Sigma, which represent the open subsets of \mathbb{Q}, rather than pairs of functions into the class of propositions. R DR_D is defined as the set of all Σ\Sigma-Dedekind cuts, or as the Σ \Sigma -Dedekind complete archimedean field. This will ensure that the set of real numbers is a set rather than a proper class, at the cost of Dedekind completeness, which always results in a proper class in constructive predicative mathematics. Nevertheless, a significant portion of real analysis could be developed using this approach, such as differential calculus, integral calculus, and differential geometry.

It is also possible to assert the existence of R\mathbf{R} by fiat, much like N\mathbf{N} exists by the axiom of infinity. This is the approach taken by the classical school; they use 𝒫N\mathcal{P}\mathbf{N} instead of R\mathbf{R} directly, but these are isomorphic by excluded middle. This is natural from the perspective of predicative set theory as a weak form of higher-order logic; you assert the existence of N\mathbf{N}, 𝒫N\mathcal{P}\mathbf{N}, and maybe 𝒫𝒫N\mathcal{P}\mathcal{P}\mathbf{N}, then stop.

There is also the question of what exactly it means to say that R\mathbf{R} exists; is it a set or a proper class? Without function sets, the distinction between these is not clear-cut; higher-order logic suggests a hierarchy of more and more proper (less and less small) classes rather than a single unified notion of set and class, which is similar to the concept of a universe in a topos for impredicative set theory. If you allow N\mathbf{N} only as a proper class, then you are basically still doing finite mathematics; if you allow R\mathbf{R} only as a proper class, then you are doing predicative mathematics.

Formalising mathematics

How much of mathematics can be done predicatively?

A surprisingly large amount of mathematics can be formalised, using various coding tricks, in a theory in which N\mathbf{N} is a set but 𝒫N\mathcal{P}\mathbf{N} is a proper class. This is somewhat easier in Nik Weaver?'s ‘conceptualist’ approach, which accepts 𝒫𝒫N\mathcal{P}\mathcal{P}\mathbf{N} as a proper class (so that 𝒫N\mathcal{P}N and \mathbb{R} are small); the encoding is not really more complicated than what is usually done in material set theory for ordered pairs and the like. Note that these are conservative over Peano arithmetic (PAPA); that is, anything expressible in PAPA and provable in these systems is provable in PAPA (which certainly cannot be said of ZFC or ETCS, which prove the consistency of PAPA).

Constructive mathematics generally requires great care with anything after the middle of the 19th century other than basic discrete mathematics, but requiring it to be predicative does not usually add much difficulty, as long as function sets are allowed. This even extends to category theory, which is not usually contemplated in the classical approach. (However, the internal logic of a Π\Pi-WW-pretopos is certainly not conservative over PAPA; it already proves consistency of the latter.)

References

Discussion of predicative toposes is in

Constructive predicative definitions of the real numbers are discussed in

Discussion

A discussion was had on this page, now archived at the nForum

Last revised on November 16, 2022 at 20:44:31. See the history of this page for a list of all contributions to it.