basic constructions:
strong axioms
further
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 $x$ as the least upper bound of a given set $A$ is impredicative, because it characterizes $x$ as a particular element of some set (the upper bounds of $A$) which includes $x$. Possibly $x$ can be defined in some other way and then proved to be the least upper bound of $A$, 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.
Not all of these axioms are rejected by all predicativists, but they at least come under some suspicion.
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.
The axiom that any set has a power set is perhaps the fundamental feature missing from predicative mathematics. In particular, the sequence
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.
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 $A$ is a set and $P$ is a function from $A$ to the set of truth values, then there is a set
To be precise, however, this $P$ 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 $P$ may be guarded by power classes.
We need more on this, particularly with regards to the classical school and replacement.
One sometimes speaks of forbidding function sets instead of power sets. That is, it is the sequence
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 $\Omega$ of truth values.
With excluded middle, the set of truth values is easy to achieve, as $\{0,1\}$; in particular, if you have $\mathbf{N}$, then you certainly have $\Omega$. 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 $CZF$ and justified by Per Martin-Löf's $ITT$.
Brouwer, on the other hand, did not accept the sequence above, although his followers differ on when (if ever) it stops.
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).
Some classical predicativists accept choice operators. This includes dependent type theories which include choice operators but reject dependent function types. But from the perspective of weakly predicative constructive mathematics, choice operators is impredicative, since it implies excluded middle, and thus power sets (given function sets) and unbounded separation (given replacement).
In predicative mathematics with choice operators, having bijection sets implies power sets, since choice operators imply the axiom of choice, which in turn implies power sets.
In the constructive school, one would sometimes have multiple sets of propositions, but those only represent the set of $U$-small propositions or subsingletons $\Omega_U$, given a universe $U$, rather than the set of all propositions $\Omega$. In general, given any two universes $U$ and $V$, one cannot prove that $\Omega_U$ is equivalent to $\Omega_V$. This is common in type theoretic models of constructive mathematics.
The axiom of propositional resizing is then statement that given any two universes $U$ and $V$, the sets $\Omega_U$ and $\Omega_V$ of propositions in $U$ and $V$ respectively are in bijection with each other $\Omega_U \cong \Omega_V$. 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.
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 $Set$. Coinductive types, which are the final coalgebras of these functors, also exist in impredicative theories, but not predicatively.
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 $\forall \alpha:Type, \alpha\to\alpha$, which has the property that an inhabitant of that type is a “function” which assigns to every type $\alpha$, an endomorphism of $\alpha$. This is clearly impredicative, since the type $\forall \alpha:Type, \alpha\to\alpha$ is also a possible value of $\alpha$.
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 $\alpha$, and thus can equally well be applied to any $\alpha$. For instance, consider the operation which assigns to every $\alpha$ the identity $id\colon \alpha\to\alpha$; this is defined in exactly the same way for every $\alpha$, and hence inhabits the type $\forall \alpha:Type, \alpha\to\alpha$.
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.
So what is the category of sets in predicative mathematics?
At bottom, let us suppose that $Set$ 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 $\mathbf{N}$, although the proofs that these exist if $\mathbf{N}$ does rely on possibly impredicative axioms. Then $Set$ is a Heyting $W$-pretopos.
If you accept function sets, then $Set$ is locally cartesian closed and thus a $\Pi$-pretopos. In this case, $\mathbf{N}$ is enough to get all $W$-types, so we have a $\Pi$-$W$-pretopos. If you accept excluded middle, then $Set$ is a Boolean pretopos or even a Boolean $W$-pretopos. But a Boolean $\Pi$-pretopos is necessarily a topos, which would make the theory impredicative.
However, $Set$ 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.
An important question in predicative mathematics is the status of the set $\mathbf{R}$ of real numbers. This set is often constructed as a subset $R_D$ of $\mathcal{P}\mathbf{N}$ or as a subquotient $R_C$ of $\mathbf{N}^{\mathbf{N}}$, neither of which can be formed in an arbitrary Heyting $W$-pretopos. The latter can be formed in a $\Pi$-$W$-pretopos, but it is not necessarily correct.
The constructive school of predicativism can construct $\mathbf{R}$ in various ways. One method is to use $R_C$ 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 ($WCC$), which is accepted by most constructive schools. Using subset collection, a variation on $R_C$ is possible which can be proved Dedekind-complete without $WCC$.
However, not all constructive predicative mathematics accept subset collection or weak countable choice. Another method is to use a predicative version of $R_D$, where given a $\sigma$-frame $\Sigma$, $\Sigma$-Dedekind cuts are defined as pairs of functions $L, U$ from the rational numbers $\mathbb{Q}$ to $\Sigma$, which represent the open subsets of $\mathbb{Q}$, rather than pairs of functions into the class of propositions. $R_D$ is defined as the set of all $\Sigma$-Dedekind cuts, or as the $\Sigma$-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 $\mathbf{R}$ by fiat, much like $\mathbf{N}$ exists by the axiom of infinity. This is the approach taken by the classical school; they use $\mathcal{P}\mathbf{N}$ instead of $\mathbf{R}$ 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 $\mathbf{N}$, $\mathcal{P}\mathbf{N}$, and maybe $\mathcal{P}\mathcal{P}\mathbf{N}$, then stop.
There is also the question of what exactly it means to say that $\mathbf{R}$ 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 $\mathbf{N}$ only as a proper class, then you are basically still doing finite mathematics; if you allow $\mathbf{R}$ only as a proper class, then you are doing predicative 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 $\mathbf{N}$ is a set but $\mathcal{P}\mathbf{N}$ is a proper class. This is somewhat easier in Nik Weaver?'s ‘conceptualist’ approach, which accepts $\mathcal{P}\mathcal{P}\mathbf{N}$ as a proper class (so that $\mathcal{P}N$ 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 ($PA$); that is, anything expressible in $PA$ and provable in these systems is provable in $PA$ (which certainly cannot be said of ZFC or ETCS, which prove the consistency of $PA$).
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 $\Pi$-$W$-pretopos is certainly not conservative over $PA$; it already proves consistency of the latter.)
Solomon Feferman; Relationships between Constructive, Predicative and Classical Systems of Analysis (PDF).
Nik Weaver; papers on conceptualism.
from the Standford Encyclopedia of Philosophy:
Discussion of predicative toposes is in
Ieke Moerdijk, Erik Palmgren, Type theories, toposes and constructive set theory: predicative aspects of algebraic set theory Ann. Pure Appl. Logic, 114(1-3):155–201, 2002
Benno van den Berg, Predicative toposes (arXiv:1207.0959)
Constructive predicative definitions of the real numbers are discussed in
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
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