Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.
The logic of the theories discussed here is constructive in that it rejects , i.e. that the disjunction automatically holds for all propositions. This requires rejection of strong choice principles and the rewording of some standard axioms. For example, the Axiom of Choice implies for the formulas in one's adopted Separation schema, by Diaconescu's theorem. Similar results hold for the Axiom of Regularity in its standard form. In turn, constructive theories often do not permit classical proofs of properties that are e.g. provably computationally undecidable. A restriction to constructive logic leads to stricter requirements regarding which characterizations of a set constitute a (mathematical, that is total) function. This is often because the predicate in a casewise would-be definition may not be decidable. Compared to the classical counterpart, one is also less likely to prove the existence of relations that cannot be realized. This then also affects the provability of statements about total orders such as that of all ordinal numbers, expressed by truth and negation of the terms in the order defining disjunction . And this in turn affects the proof theoretic strength defined in ordinal analysis. That said, theories without tend to prove classically equivalent reformulations of classical theorems. For example, in Constructive analysis one cannot prove the intermediate value theorem in its textbook formulation, but one can prove theorems with algorithmic content that, as soon as is assumed, are at once classically equivalent to the classical statement. The difference is that the constructive proofs are harder to find.
Many theories studied in constructive set theory are mere restrictions of Zermelo–Fraenkel set theory () with respect to their axiom as well as their underlying logic. Such theories can then also be interpreted in any model of . As far as constructive realizations go there is a realizability theory and Aczel's theory constructive Zermelo-Fraenkel () has been interpreted in a Martin-Löf type theories, as described below. In this way, set theory theorems provable in and weaker theories are candidates for a computer realization. More recently, presheaf models for constructive set theories have been introduced. These are analogous to unpublished Presheaf models for intuitionistic set theory developed by Dana Scott in the 1980s.
The subject of constructive set theory (often "") begun by John Myhill's work on the theory also called , a theory of several sorts and bounded quantification, aiming to provide a formal foundation for Errett Bishop's program of constructive mathematics. Below we list a sequence of theories in the same language as , leading up to Peter Aczel's well studied , and beyond. is also characterized by the two features present also in Myhill's theory: On the one hand, it is using the Predicative Separation instead of the full, unbounded Separation schema, see also Lévy hierarchy. Boundedness can be handled as a syntactic property or, alternatively, the theories can be conservatively extended with a higher boundedness predicate and its axioms. Secondly, the impredicative Powerset axiom is discarded, generally in favor of related but weaker axioms. The strong form is very casually used in classical general topology. Adding to a theory even weaker than recovers , as detailed below. The system, which has come to be known as Intuitionistic Zermelo–Fraenkel set theory (), is a strong set theory without . It is similar to , but less conservative or predicative. The theory denoted is the constructive version of , the classical Kripke–Platek set theory where even the Axiom of Collection is bounded.
Subtheories of ZF
This article focuses on logical implications of the most basic constructive set theory, as opposed to e.g. contemporary ordinal analysis therein.
In the following main section we discuss common axiom candidates which make for frameworks in which all proofs are also proofs in .
Below we use the greek as a predicate variable in axiom schemas and use or for particular predicates. Unique existence e.g. means . Quantifiers range over set and those are denoted by lower case letters.
As is common in the study of set theories, one makes use set builder notation for classes, which, in most contexts, are not part of the object language but used for concise discussion. In particular, one may introduce notation declarations of the corresponding class via "", for the purpose of expressing as . Logically equivalent predicates can be used to introduce the same class. One also writes as shorthand for .
As is common, we may abbreviate by and express the subclass claim , i.e. , by .
For a property , trivially . And so follows that .
We start with axioms that are virtually always deemed uncontroversial and part of all theories considered in this article.
Denote by the statement expressing that two classes have exactly the same elements, i.e. , or equivalently .
The following axiom gives a means to prove equality "" of two sets, so that through substitution, any predicate about translates to one of .
By the logical properties of equality, the converse direction holds automatically.
Note that in a constructive interpretation, the elements of a subclass of may come equipped with more information than those of , in the sense that being able to judge is being able to judge . And (unless the whole disjunction follows from axioms) in the Brouwer–Heyting–Kolmogorov interpretation, this means to have proven or having rejected it. As may be not decidable for all elements in , the two classes must a priori be distinguished.
Consider a property that provably holds for all elements of a set , so that , and assume that the class on the left hand side is established to be a set. Note that, even if this set on the left informally also ties to proof-relevant information about the validity of for all the elements, the Extensionality axiom postulates that, in our set theory, the set on the left hand side is judged equal to the one on the right hand side.
Modern type theories may instead aim at defining the demanded equivalence "" in terms of functions, see e.g. type equivalence. The related concept of function extensionality is often not adopted in type theory.
Other frameworks for constructive mathematics might instead demand a particular rule for equality or apartness come for the elements of each and every set discussed. Even then, the above definition can be used to characterize equality of subsets and .
Two other basic axioms are
The two axioms may also be formulated stronger in terms of "", e.g. in the context of this is not necessary.
Together, these two axioms imply the existence of the binary union of two classes and when they have been established to be sets, and this is denoted or . Define class notation for finite elements via disjunctions, e.g., says , and define the successor set as . A sort of blend between pairing and union, an axiom more readily related to the successor is the Axiom of adjunction. It is relevant for the standard modeling of individual Neumann ordinals. This axiom would also readily be accepted, but is not relevant in the context of stronger axioms below. Denote by the standard ordered pair model .
The property that is false for any set corresponds to the empty class, denoted by or zero, 0. That this is a set readily follows from other axioms, such as the Axiom of Infinity below. But if, e.g., one is explicitly interested in excluding infinite sets in one's study, one may at this point adopt the Axiom of empty set
In the following we will make use of axiom schemas, i.e. we postulate axioms for some collection of predicates. Note that some of the stated axiom schemas are often presented with set parameters as well, i.e. variants with extra universal closures such that the 's may depend on the parameters.
Basic constructive set theory consists of several axioms also part of standard set theory, except the Separation axiom is weakened. Beyond the three axioms above, it adopts the
The axiom amounts to postulating the existence of a set obtained by the intersection of any set and any predicatively described class . When the predicate is taken as for proven to be a set, one obtains the binary intersection of sets and writes .
The schema is also called Bounded Separation, as in Separation for set-bounded quantifiers only. It is the axiom schema that makes reference to syntactic aspects of predicates. The bounded formulas are also denoted by in the set theoretical Lévy hierarchy, in analogy to in the arithmetical hierarchy. (Note however that the arithmetic classification is sometimes expressed not syntactically but in terms of subclasses of the naturals, and also note that there the bottom level has several common definitions, some not allowing the use of some total functions. The distinction is not relevant on the level or higher.) The restriction in the axiom is also gatekeeping impredicative definitions. For example, without the Axiom of power set, one should not expect a class defined as to be a set, where denotes some 2-ary predicate. Note that if this subclass is provably a set, then the term thus defined is also in the scope of term variable used to define it. While, in this way, predicative Separation leads to fewer given class definitions being sets, it must be emphasized that many class definitions that are classically equivalent are not so when restricting oneself to constructive logic. Thus, in this way, one gets a richer theory of sets constructively. Due to the potential undecidability of general predicates, the notion of subclass is more elaborate in constructive set theories than in classical ones, as we will see.
As noted, from Separation and the existence of any set (e.g. Infinity below) and the predicate that is false of any set will follow the existence of the empty set.
Next, we consider the
Axiom schema of Replacement: For any predicate ,
It is granting existence, as sets, of the range of function-like predicates, obtained via their domains.
Replacement and the axiom of Set Induction (introduced below) suffices to axiomize hereditarily finite sets constructively and that theory is also studied without Infinity. For comparison, consider the very weak classical theory called General set theory that interprets the class of natural numbers and their arithmetic via just Extensionality, Adjunction and full Separation.
In , Replacement is mostly important to prove the existence of sets of high rank, namely via instances of the axiom schema where relates relatively small set to bigger ones, .
Constructive set theories commonly have Axiom schema of Replacement, sometimes restricted to bounded formulas. However, when other axioms are dropped, this schema is actually often strengthened - not beyond , but instead merely to gain back some provability strength.
Only when assuming does Replacement already imply full Separation. Within this conservative context of , the Bounded Separation schema is actually equivalent to Empty Set plus the existence of the binary intersection for any two sets. The latter variant of axiomatization does not make use of a schema.
Denote by the Inductive property, e.g. . In terms of a predicate underling the class, this would be translated as . Note that denotes a generic set variable here. Write for . Define a class .
For some fixed predicate , the statement expresses that is the smallest set among all sets for which holds true. The elementary constructive Set Theory has the axiom of as well as
The second universally quantified conjunct expresses mathematical induction for all in the universe of discourse, i.e. for sets, resp. for predicates if they indeed define sets . In this way, the principles discussed in this section give means of proving that some predicates hold at least for all elements of . Be aware that even the quite strong axiom of full mathematical induction (induction for any predicate, discussed below) may also be adopted and used without ever postulating that forms a set.
Weak forms of axioms of infinity can be formulated, all postulating that some set with the common natural number properties exist. Then full Separation may be used to obtain the "sparse" such set, the set of natural numbers. In the context of otherwise weaker axiom systems, an axiom of infinity should be strengthened so as to imply existence of such a sparse set on its own. One weaker form of Infinity reads
which can also be written more concisely using . The set thus postulated to exist is generally denoted by , the smallest infinite von Neumann ordinal. For elements of this set, the claim is decidable.
With this, proves induction for all predicates given by bounded formulas. The two of the five Peano axioms regarding zero and one regarding closedness of with respect to follow fairly directly from the axioms of infinity. Finally, can be proven to be an injective operation.
The basic order is captured by membership in this model. For the sake of standard notation, let denote an initial segment of the natural numbers, for any , including the empty set.
Naturally, the logical meaning of existence claims is a topic of interest in intuitionistic logic. Here we focus on total relations.
NB: The proof calculus, in a constructive mathematical framework, of statements such as
might be set up in terms of programs on represented domains and possibly having to witness the claim. This is to be understood in the sense of, informally speaking, , where denotes the value of a program as mentioned, but this gets into questions of realizability theory. For a stronger context, if and when the proposition holds, then demanding that this shall always be possible with a realized by some total recursive function is a possible Church's thesis postulate adopted in, consequently, strictly non-classical Russian constructivism. In the previous paragraph, "function" needs to be understood in the computable sense of recursion theory - this occasional ambiguity must also be watched out for below.
Relatedly, consider Robinson arithmetic, which is a classical arithmetic theory that substitutes the full mathematical induction schema for a predecessor number existence claim. It is a theorem that that theory represents exactly the recursive functions in the sense of defining predicates that are provably a total functional relations,
Now in the current set theoretical approach, we define the property involving the function application brackets, , as and speak of a function when provably
which notably involves an existential quantifier. Whether a subclass can be judged to be a function will depend on the strength of the theory, which is to say the axioms we adopt.
Note that a general class could fulfill the above predicate without being a subclass of the product , i.e. the property is expressing not more or less than functionality w.r.t. inputs from . Care must be taken with nomenclature "function", which use in most mathematical frameworks, also because some tie a function term itself to a particular codomain. Variants of the functional predicate definition using apartness relations on setoids have been defined as well.
Let (also written ) denote the class of such set functions. Using the standard class terminology, one can freely make use of functions, given their domain is a set. The functions as a whole will be sets if their codomain is. When functions are understood as just function graphs as above, the membership proposition is also written . Below might write for for the sake of distinguishing it from ordinal exponentiation.
Separation lets use cut out subsets of products , at least when they are described in a bounded fashion. Write for . Given any , we are now led to reason about classes such as
The boolean-valued characteristic functions are among such classes. But be aware that may in generally not be decidable. That is to say, in absence of any non-constructive axioms, the disjunction may not be provable, since we require an explicit proof of either disjunct. When
cannot be witnessed for all , or uniqueness of a term not be proven, then we cant constructively judge the comprehended collection to be functional.
For and any natural , have
So in classical set theory, where, by , the propositions in the definition is decidable, so is subclass membership. If the set is not finite, successively "listing" all numbers in by simply skipping those with classically constitutes an increasing surjective sequence . There, we can obtain a bijective function. In this way, the classical class of functions is provably rich, as it also contains objects that are beyond what we know to be effectively computable, or programmatically listable in praxis.
In contrast, for reference, in computability theory, the computable sets are ranges of non-decreasing total functions (in the recursive sense), at the level of the arithmetical hierarchy, and not higher. Deciding a predicate at that level amounts to solving the task of eventually finding a certificate that either validates or rejects membership. As not every predicate is computably decidable, the theory alone will not claim (prove) that all infinite are the range of some bijective function with domain .
To be finite means there is a bijective function to a natural. To be subfinite means to be a subset of a finite set. The claim that being a finite set is equivalent to being subfinite is equivalent to .
But being compatible with , the development in this section also always permits "function on " to be interpreted as an object not necessarily given as lawlike sequence. Applications may be found in the common models for claims about probability, e.g. statements involving the notion of "being given" an unending random sequence of coin flips.
- Axiom of countable choice: If , we can form the one-to-many relation-set . The axiom of countable choice would grant that whenever , we can form a function mapping each number to a unique value.
- Axiom of dependent choice: It is implied by the more general axiom of dependent choice, extracting a function from any entire relation on an inhabited set. Countable choice is akin to a case of the constructive Church's thesis and indeed dependent choice holds in many realizability models and it is thus adopted in many constructive schools.
- Relativized dependent choice: The stronger relativized dependent choice principle is a variant of it - a schema involving an extra predicate variable. Adopting this for just bounded formulas in , the theory already proves the -induction, further discussed below.
- Axiom of choice: Dependent choice is implied by the axiom of choice concerning functions on general domains.
We conclude by a remark to highlight the strength of choice and its relation to matters of Intentionality. So consider the subfinite classes
and assume they are proven sets. Here the full Axiom of Choice, granting existence of a map with into distinguishable elements (equality on the codomain is decidable), implies that . So the existence claim of general choice functions is non-constructive. To better understand this phenomenon, consider cases of logical implications, such as et cetera. Take note that the sets and are as contingent as the predicates involved in their definition. The difference between the discrete codomain of some natural numbers and the domain lies in the fact that a priori little is known about the latter. It is the case that and , regardless of , making a contender. But in the case of , as implied by provability of , we have so that there is extensionally only one possible function input to a choice choice function, now just . So when considering the functional assignment , then unconditionally declaring would not be consistent. In the described way, the Axiom of Separation ties predicates to set equality, and in turn to information about functions. Choice may be not adopted in an otherwise strong set theory, because the mere claim of function existence does not realize a particular function.
We proceed here in a fashion agnostic to the discussed choice principles.
For reference, on the classical arithmetic side, the theories starting with bounded arithmetic adopt different conservative induction schemata and may adds symbols for particular functions, leading to theories between Robinson- and Peano arithmetic. Despite being classical, most of such theories are however relatively weak regarding proof of totality for some more fast growing functions or regarding the existence of proof theoretic ordinals below . Some of the most basic examples include elementary function arithmetic , with ordinal . The -induction schema, as e.g. implied by the relativized dependent choice, means induction for those subclasses of naturals computable via a finite search with unbound (finite) runtime. The schema is also equivalent to -induction schema. It is also adopted in the relatively weak classical first-order arithmetic or the classical second-order reverse mathematics base system . That second-order theory is -conservative over primitive recursive arithmetic , so proves all primitive recursive functions total, and is relevant in this discussion insofar as all sets of naturals it proves to exist are also computable sets. Those last mentioned arithmetic theories all have ordinal .
However, does not interpret full primitive recursion. Indeed, despite having the Replacement axiom, the theory still does not proof the addition function to be a set function. On the other hand, many statements can be proven per individual set in this theory (as opposed to expressions involving a universal quantifier, as e.g. available with an induction principle) and objects of mathematical interest can be made use of at the class level on an individual basis. As such, the axioms listed so far suffice as basic working theory for a good portion of basic mathematics.
Going beyond with regards to arithmetic, the axiom granting definition of set functions via iteration-step set functions must be added. What is necessary is the set theoretical equivalent of a natural numbers object. This then enables an interpretation of Heyting arithmetic, . With this, arithmetic of rational numbers can then also be defined and its properties, like uniqueness and countability, be proven.
A set theory with this will also prove that, for any naturals and , the function spaces
are sets. Conversely, a proof of the seeked iteration principle may be based on the collection of functions we would want to write as and the existence of this is implied by assuming that the individual function spaces on finite domains into sets form sets themselves. This remark should motivate the adoption of an axiom of more set theoretical flavor, instead of just directly embedding arithmetic principles into our theory. The induction principle obtained through the next, more set theoretical axiom will, however, still not prove the full mathematical induction schema. But it will validate the claim used in the above example directly.
We already adopted a weakened form of the Separation schema, and more of the standard axioms shall be weakened for a more predicative and constructive theory. The first one of those is the Powerset axiom, which, in effect, we adopt for decidable subsets of the theory.
The characterization of the class of all subsets of a set involves unbounded universal quantification, namely . Here has been defined in terms of the membership predicate above. The statement itself is . So in a mathematical set theory framework, the power class is defined not in a bottom-up construction from its constituents (like an algorithm on a list, that e.g. maps ) but via a comprehension over all sets.
The richness of the powerclass in a theory without excluded middle can best be understood by considering small classically finite sets. For any formula , the class equals 0 when can be rejected and 1 when can be proven, but may also not be decidable at all. In this view, the powerclass of the singleton , i.e. the class , or maybe suggestively "", and usually denoted by , is called the truth value algebra. The -valued functions on a set inject into and thus correspond to its decidable subsets.
So we will next consider the axiom .
In words, the axioms says that given two sets , the class of all functions is, in fact, also a set. This is certainly required, for example, to formalize the object map of an internal hom-functor like . The use of the Exponentiation axioms derives from the fact that function spaces being sets means quantification over their functions is a bounded notion, enabling use of Separation. It has notable implications: Note that adopting it means that quantification over the elements of certain classes of functions becomes a bounded notion, also when the function spaces are even classically uncountable. E.g. the collection of all functions , i.e. the set of points underlying the Cantor space, is uncountable, by Cantor's diagonal argument, and can at best be taken to be subcountable. (In this section we start to use the symbol for the semiring of natural numbers in expressions like or write just to avoid conflation of cardinal- with ordinal exponentiation.)
The set theory with Exponentiation now also proves the existence of any primitive recursive function on the naturals , as set functions in the set . Relatedly, we now obtain the ordinal-exponentiated number , which may be characterized as . Spoken more generally, Exponentiation proves the union all finite sequences over a countable set to be a countable set. And indeed, unions of the ranges of any countable family of counting functions is countable.
As far as comprehension goes, the theory now proves the collection of all the countable subsets of any set (the collection is a subclass of the powerclass) to be a set. Also the pigeon hole principle can be proven.
Coming back to the original power class consideration: When assuming that all formulas are decidable, i.e. assuming , one can show not only that becomes a set, but more concretely that it is this two-element set. Assuming for bounded formulas, Separation lets one demonstrate that any powerclass is a set. Alternatively, full Powerset is equivalent to merely assuming that the class of all subsets of forms a set. Full Separation is equivalent to assuming that each individual subclass of is a set.
NB: Restricting function spaces
In the following remark function and claims made about them is again meant in the sense of computability theory. The μ operator enables all partial general recursive functions (or programs, in the sense that they are Turing computable), including ones e.g. non-primitive but total, such as the Ackermann function. The definition of the operator involves predicates over the naturals and so the theoretical analysis of functions and their totality depends on the formal framework at hand. Either way, those natural numbers that are, in computability theory, thought of as indices for the computable functions which are total, are , in the arithmetical hierarchy. Which is to say it is still a subclass of the naturals. And there, totality, as a predicate on all programs, is famously computably undecidable.
A non-classical constructive Church's thesis, as per assumption in its antecedent, concerns the predicate definitions (and thus here set functions) that are demonstrably total and it postulates they corresponds to computable programs. Adopting the postulate makes into a "sparse" set, as viewed from classical set theory. See subcountability.
Category and type theoretic notions
So in this context, function spaces are more accessible than classes of subsets, as is the case with exponential objects resp. subobjects in category theory. In category theoretical terms, the theory essentially corresponds to constructively well-pointed Cartesian closed Heyting pretoposes with (whenever Infinity is adopted) a natural numbers object. Existence of powerset is what would turn a Heyting pretopos into an elementary topos. Every such topos that interprets is of course a model of these weaker theories, but locally Cartesian closed pretoposes have been defined that e.g. interpret but reject full Separation and Powerset.
In type theory, the expression "" exists on its own and denotes function spaces, a primitive notion. These classes naturally appear, for example, as the type of the currying bijection between and , an adjunction. A typical type theory with general programming capability - and certainly those that can model , which is considered a constructive set theory - will have a type of integers and function spaces representing , and as such also include types that are not countable. This just implies and means that among the function terms , none have the property of being a bijection.
Constructive set theories are also studied in the context of applicative axioms.
Towards the reals
As mentioned, Exponentiation implies recursion principles and so in , one can reason about sequences or about shrinking intervals in and this also enables speaking of Cauchy sequences and their arithmetic. Any Cauchy real number is a collection of sequences, i.e. subset of a set of functions on . More axioms are required to always grant completeness of equivalence classes of such sequences and strong principles need to be postulated to imply the existence of a modulus of convergence for all Cauchy sequences. Weak countable choice is generally the context for proving uniqueness of the Cauchy reals as complete (pseudo-)ordered field. "Pseudo-" here highlights that the order will, in any case, constructively not always be decidable.
As in the classical theory, Dedekind cuts are characterized using subsets of algebraic structures such as : The properties of being inhabited, numerically bounded above, "closed downwards" and "open upwards" are all bounded formulas with respect to the given set underlying the algebraic structure. A standard example of a cut, the first component indeed exhibiting these properties, is the representation of given by
(Depending on the convention for cuts, either of the two parts or neither, like here, may makes use of the sign .)
The theory given by the axioms so far validates that a pseudo-ordered field that is also Archimedean and Dedekind complete, if it exists at all, is in this way characterized uniquely, up to isomorphism. However, the existence of just function spaces such as does not grant to be a set, and so neither is the class of all subsets of that do fulfill the named properties. What is required for the class of Dedekind reals to be a set is an axiom regarding existence of a set of subsets.
In either case, fewer statements about the arithmetic of the reals are decidable, compared to the classical theory.
NB: Constructive schools
Non-constructive claims valuable in the study of constructive analysis are commonly formulated as concerning all binary sequences, i.e. functions . That is to say claims quantified by . Maybe a most prominent example is the limited principle of omniscience , postulating a disjunctive property, like does, for functions. (Example functions can be constructed in raw such that, if is consistent, the competing disjuncts are -unprovable.) In a constructive analysis context with countable choice, is e.g. equivalent to the claim that every real is either rational or irrational - again without the requirement to witness either disjunct. The principle is independent of e.g. .
For some propositions employed in theories of constructive analysis that are not provable using just base intuitionistic logic, see the Markov's principle or even the non-classical constructive Church's thesis on the recursive mathematics side, and as well as Kripke's schema (turning all subclasses of countable), bar induction, the fan theorem or even the non-classical continuity principle determining functions on unending sequences through finite initial segments on Brouwerian intuitionist side.
In previous sections, bounded Separation already established the validity of induction for bounded definitions. In set language, induction principles can read with the antecedent defined as further above. It is instructive to note that a proposition in the consequent, like , here expressed using class notation involving a subclass that may not constitute a set - meaning many axioms wont apply - and the plain are just two ways of formulating the same desired claim (an -indexed conjunction of claims here, in particular.) So a set theoretical framework with just bounded Separation can be strengthened through arithmetical induction schemas for unbounded predicates. It is however worth noting that in the program of predicative arithmetic, the mathematical induction schema has been criticized as possibly being impredicative, when natural numbers are defined as the object which fulfill this schema.
The iteration principle for set functions mentioned before is, alternatively to Exponentiation, also implied by the full induction schema over one's structure modeling the naturals (e.g. ). It is often formulated directly in terms of predicates, as follows. We consider schema -:
Axiom schema of full mathematical induction: For any predicate on ,
Here the 0 denotes as above, and the set denotes the successor set of , with . By Axiom of Infinity above, it is again a member of .
To proof the existence of a transitive closure for every set with respect to , at least a bounded iteration schema is needed.
As noted in the section on Choice, induction principles are also implied by various forms of choice principles. The full induction schema is implied by the full Separation schema.
Full Set Induction in proves full mathematical induction over the natural numbers. Indeed, it gives induction on ordinals and ordinal arithmetic. Replacement is not required to prove induction over the set of naturals, but it is for their arithmetic modeled within the set theory.
The stronger axiom - then reads as follows:
Axiom schema of Set induction: For any predicate ,
Note that holds trivially and corresponds to the "bottom case" in the standard framework. The variant of the Axiom just for bounded formulas is also studied independently and may be derived from other axioms.
The axiom allows definitions of class functions by transfinite recursion. The study of the various principles that grant set definitions by induction, i.e. inductive definitions, is a main topic in the context of constructive set theory and their comparatively weak strengths. This also holds for their counterparts in type theory.
The Axiom of Regularity together with bounded/unbounded Separation implies Set Induction but also bounded/unbounded , so Regularity is non-constructive. Conversely, together with Set Induction implies Regularity.
This now covers variants of all of the eight Zermelo–Fraenkel axioms. Extensionality, Pairing, Union and Replacement are indeed identical. Infinity is stated in a strong formulation and implies Emty Set, as in the classical case. Separation, classically stated redundantly, is constructively not implied by Replacement. Without the Law of Excluded Middle, the theory here is lacking, in its classical form, full Separation, Powerset as well as Regularity.
The theory does not exceed the consistency strength of Heyting arithmetic but adding at this stage would lead to a theory beyond the strength of typical type theory: Assuming Separation in unrestricted form, then adding to gives a theory proving the same theorems as minus Regularity! Thus, adding Separation and Regularity to that framework gives full and adding Choice to it gives .
The added proof-theoretical strength attained with Induction in the constructive context is significant, even if dropping Regularity in the context of does not reduce the proof-theoretical strength. Note that Aczel was also one of the main developers or Non-well-founded set theory, which rejects this last axiom.
With all the weakened axioms of and now going beyond those axioms also seen in Myhill's typed approach, the theory called (a theory with Exponentiation) strengthens the collection schema. It concerns a property for relations, giving rise to a somewhat repetitive format in its first-order formulation.
Axiom schema of Strong Collection: For any predicate ,
It states that if is a relation between sets which is total over a certain domain set (that is, it has at least one image value for every element in the domain), then there exists a set which contains at least one image under of every element of the domain. And this formulation then moreover states that only such images are elements of that codomain set. The last clause makes the axiom - in this constructive context - stronger than the standard formulation of Collection. It is guaranteeing that does not overshoot the codomain of and thus the axiom is expressing some power of a Separation procedure.
The axiom is an alternative to the Replacement schema and indeed supersedes it, due to not requiring the binary relation definition to be functional.
As a rule, questions of moderate cardinality are more subtle in a constructive setting. As arithmetic is well available in , the theory has dependent products, proves that the class of all subsets of natural numbers cannot be subcountable and also proves that countable unions of function spaces of countable sets remain countable.
This theory without , unbounded separation and "naive" Power set enjoys various nice properties. For example, it has the Existence Property: If, for any property , the theory proves that a set exist that has that property, i.e. if the theory proves the statement , then there is also a property that uniquely describes such a set instance. I.e., the theory then also proves . This can be compared to Heyting arithmetic where theorems are realized by concrete natural numbers and have these properties. In set theory, the role is played by defined sets. For contrast, recall that in , the Axiom of Choice implies the Well-ordering theorem, so that total orderings with least element for sets like are formally proven to exist, even if provably no such ordering can be described.
One may approach Power set further without losing a type theoretical interpretation. The theory known as is plus a stronger form of Exponentiation. It is by adopting the following alternative, which can again be seen as a constructive version of the Power set axiom:
Axiom schema of Subset Collection: For any predicate ,
This Subset Collection axiom schema is equivalent to a single and somewhat clearer alternative Axiom of Fullness. To this end, let is the class of all total relations between a and b, this class is given as
With this, we can state , an alternative to Subset Collection. It guarantees that there exists at least some set holding the a good amount of the desired relations. More concretely, between any two sets and , there is a set which contains a total sub-relation for any total relation from to .
- Axiom of Fullness:
Fullness implies the binary refinement property necessary to prove that the class of Dedekind cuts is a set. This does not require Induction or Collection.
This theory lacks the existence property due to the Schema, but in 1977 Aczel showed that can still be interpreted in Martin-Löf type theory, (using the propositions-as-types approach) providing what is now seen a standard model of in type theory. This is done in terms of images of its functions as well as a fairly direct constructive and predicative justification, while retaining the language of set theory. This subcountable model validates many choice principles. With a type theoretical model, has modest proof theoretic strength, see : Bachmann–Howard ordinal.
NB: Breaking with ZF
One may further add the non-classical claim that all sets are subcountable as an axiom. Then is a set (by Infinity and Exponentiation) while the class or even is provably not a set, by Cantor's diagonal argument. So this theory then logically rejects Powerset and .
In 1989 Ingrid Lindström showed that non-well-founded sets obtained by replacing the equivalent of the Axiom of Foundation (Induction) in with Aczel's anti-foundation axiom () can also be interpreted in Martin-Löf type theory.
Here, in place of the Axiom schema of replacement, we may use the
Axiom schema of collection: For any predicate ,
While the axiom of replacement requires the relation to be functional over the set (as in, for every in there is associated exactly one ), the Axiom of Collection does not. It merely requires there be associated at least one , and it asserts the existence of a set which collects at least one such for each such . together with the Collection implies Replacement.
As such, can be seen as the most straight forward variant of without .
The theory is consistent with being subcountable as well as with Church's thesis for number theoretic functions. But, as implied above, the subcountability property cannot be adopted for all sets, given the theory proves to be a set.
Changing the Axiom schema of Replacement to the Axiom schema of Collection, the resulting theory has the Numerical Existence Property.
Even without , the proof theoretic strength of equals that of .
While is based on intuitionistic rather than classical logic, it is considered impredicative. It allows formation of sets using the Axiom of Separation with any proposition, including ones which contain quantifiers which are not bounded. Thus new sets can be formed in terms of the universe of all sets. Additionally the power set axiom implies the existence of a set of truth values. In the presence of excluded middle, this set exists and has two elements. In the absence of it, the set of truth values is also considered impredicative.
In 1973, John Myhill proposed a system of set theory based on intuitionistic logic taking the most common foundation, , and throwing away the Axiom of choice and the law of the excluded middle, leaving everything else as is. However, different forms of some of the axioms which are equivalent in the classical setting are inequivalent in the constructive setting, and some forms imply . In those cases, the intuitionistically weaker formulations were then adopted for the constructive set theory.
Again on the weaker end, as with its historical counterpart Zermelo set theory, one may denote by the intuitionistic theory set up like but without Replacement, Collection or Induction.
Let us mention another very weak theory that has been investigated, namely Intuitionistic (or constructive) Kripke–Platek set theory . The theory has not only Separation but also Collection restricted, i.e. it is similar to but with Induction instead of full Replacement. It is especially weak when studied without Infinity. The theory does not fit into the hierarchy as presented above, simply because it has Axiom schema of Set Induction from the start. This enables theorems involving the class of ordinals.
Constructive set theory
- The usual Axiom of Extensionality for sets, as well as one for functions, and the usual Axiom of union.
- The Axiom of restricted, or predicative, separation, which is a weakened form of the Separation axiom from classical set theory, requiring that any quantifications be bounded to another set, as discussed.
- A form of the Axiom of Infinity asserting that the collection of natural numbers (for which he introduces a constant ) is in fact a set.
- The axiom of Exponentiation, asserting that for any two sets, there is a third set which contains all (and only) the functions whose domain is the first set, and whose range is the second set. This is a greatly weakened form of the Axiom of power set in classical set theory, to which Myhill, among others, objected on the grounds of its impredicativity.
- The usual Peano axioms for natural numbers.
- Axioms asserting that the domain and range of a function are both sets. Additionally, an Axiom of non-choice asserts the existence of a choice function in cases where the choice is already made. Together these act like the usual Replacement axiom in classical set theory.
We can roughly identify the strength of this theory with a constructive subtheories of when comparing with the previous sections.
And finally the theory adopts
Bishop style set theory
Set theory in the flavor of Errett Bishop's constructivist school mirrors that of Myhill, but is set up in a way that sets come equipped with relations that govern their discreteness. Commonly, Dependent Choice is adopted.
Not all formal logic theories of sets need to axiomize the binary membership predicate "" directly. And an Elementary Theory of the Categories Of Set (), e.g. capturing pairs of composable mappings between objects, can also be expressed with a constructive background logic (). Category theory can be set up as a theory of arrows and objects, although first-order axiomatizations only in terms of arrows are possible.
Good models of constructive set theories in category theory are the pretoposes mentioned in the Exponentiation section - possibly also requiring enough projectives, an axiom about surjective "presentations" of set, implying Countable Dependent Choice.
- Constructive mathematics
- Intuitionistic type theory
- Ordinal analysis
- Existence Property
- Law of excluded middle
- Gambino, N. (2005). "PRESHEAF MODELS FOR CONSTRUCTIVE SET THEORIES" (PDF). In Laura Crosilla and Peter Schuster (ed.). From Sets and Types to Topology and Analysis (PDF). pp. 62–96. doi:10.1093/acprof:oso/9780198566519.003.0004. ISBN 9780198566519.
- Scott, D. S. (1985). Category-theoretic models for Intuitionistic Set Theory. Manuscript slides of a talk given at Carnegie-Mellon University
- Peter Aczel and Michael Rathjen, Notes on Constructive Set Theory, Reports Institut Mittag-Leffler, Mathematical Logic - 2000/2001, No. 40
- Aczel, Peter: 1978. The type theoretic interpretation of constructive set theory. In: A. MacIntyre et al. (eds.), Logic Colloquium '77, Amsterdam: North-Holland, 55–66.
- Rathjen, M. (2004), "Predicativity, Circularity, and Anti-Foundation" (PDF), in Link, Godehard (ed.), One Hundred Years of Russell ́s Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, ISBN 978-3-11-019968-0
- Lindström, Ingrid: 1989. A construction of non-well-founded sets within Martin-Löf type theory. Journal of Symbolic Logic 54: 57–64.
- Myhill, "Some properties of Intuitionistic Zermelo-Fraenkel set theory", Proceedings of the 1971 Cambridge Summer School in Mathematical Logic (Lecture Notes in Mathematics 337) (1973) pp 206-231
- Troelstra, Anne; van Dalen, Dirk (1988). Constructivism in Mathematics, Vol. 2. Studies in Logic and the Foundations of Mathematics. p. 619. ISBN 978-0-444-70358-3.
- Aczel, P. and Rathjen, M. (2001). Notes on constructive set theory. Technical Report 40, 2000/2001. Mittag-Leffler Institute, Sweden.