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Class (set theory)
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In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see § Paradoxes). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong.

Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology.[1] Many discussions of "classes" in the 19th century and earlier are really referring to sets, or rather perhaps take place without considering that certain classes can fail to be sets.[non-primary source needed]

Examples

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The collection of all algebraic structures of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.

The surreal numbers are a proper class of objects that have the properties of a field.

Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers.

One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice on three or more generators.

Paradoxes

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The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate, on the other hand, can have proper classes as members.[2]

Classes in formal set theories

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ZF set theory does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes.[3] For example, one can reduce the formula to . For a class and a set variable symbol , it is necessary to be able to expand each of the formulas , , , and into a formula without an occurrence of a class.[4]p. 339

Semantically, in a metalanguage, the classes can be described as equivalence classes of logical formulas: If is a structure interpreting ZF, then the object language "class-builder expression" is interpreted in by the collection of all the elements from the domain of on which holds; thus, the class can be described as the set of all predicates equivalent to (which includes itself). In particular, one can identify the "class of all sets" with the set of all predicates equivalent to .[citation needed]

Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".

In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula with the property that for any set there is no more than one set such that the pair satisfies . For example, the class function mapping each set to its powerset may be expressed as the formula . The fact that the ordered pair satisfies may be expressed with the shorthand notation .

Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension of ZFC.

Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZFC.

In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.

Notes

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References

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Class (set theory)

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In set theory, a class is a collection of sets defined by a in the of , serving as an extensional totality that may or may not itself be a set; classes that are not sets are termed proper classes, such as the class of all sets (denoted VV) or the class of all ordinals (denoted ONON). Proper classes cannot be elements of any other class due to their size, which prevents paradoxes like from arising within the . Classes were formalized in the von Neumann–Bernays–Gödel (NBG) set theory, developed in the early by , Paul Bernays, and to extend (ZF) by treating classes as primitive objects alongside sets. In NBG, the axiom schema of class comprehension asserts that for any formula ϕ(x)\phi(x) with set parameters, there exists a class {x:ϕ(x)}\{x : \phi(x)\}, while the distinction between sets and proper classes is captured by the axiom that a class is a set if and only if it is an element of some class. This framework includes a global axiom of choice applying to all classes and is finitely axiomatizable, unlike the infinite schema in ZF. NBG is a conservative extension of (ZFC), meaning it proves exactly the same theorems about sets as ZFC but allows direct quantification over proper classes, simplifying arguments in , , and forcing techniques. In contrast to ZFC, where classes are merely syntactic abbreviations for formulas without formal existence, NBG's explicit treatment of classes enables reasoning about the entire set-theoretic universe, such as the von Neumann VV, and supports applications in axioms and independence proofs.

Background and Motivation

Historical Development

The concept of classes in set theory emerged as a response to the foundational challenges posed by Georg Cantor's naive set theory, which he developed in the late through works such as his 1895–1897 Beiträge zur Begründung der transfiniten Mengenlehre. Cantor's unrestricted comprehension principle allowed for the formation of sets from any property, leading to a crisis around 1900 when paradoxes, including Cantor's own concerning the set of all sets, revealed inconsistencies in this approach. This crisis intensified with Bertrand Russell's identification of what became known as in a 1902 letter to , published in 1903, which demonstrated that the set of all sets not containing themselves leads to a contradiction and prompted early distinctions between collectible sets and broader classes. Ernst Zermelo addressed these issues in 1908 by axiomatizing in his paper "Untersuchungen über die Grundlagen der Mengenlehre I," introducing that restricted set formation to avoid paradoxes while focusing solely on sets, though the "universe of all sets" remained informal. Abraham Fraenkel extended this in 1922 with refinements, including the axiom of replacement, further solidifying Zermelo-Fraenkel (ZF) but still without formalizing classes. John von Neumann introduced the explicit concept of classes in 1925 in his axiomatization, treating classes as definable collections that may be too large to be sets, providing a framework to handle proper classes like the universe of all sets. This approach was formalized and simplified in von Neumann–Bernays–Gödel (NBG) set theory, with Gödel's 1940 monograph "The Consistency of the and of the Generalized Continuum-Hypothesis" presenting a conservative extension of ZF that incorporates classes rigorously.

Need for Classes in Set Theory

In pure , such as Zermelo-Fraenkel set theory with the (ZFC), the collection of all sets, often denoted as VV, cannot be treated as a set itself, as doing so leads to contradictions. For instance, if VV were a set, then by the of separation, one could form the {x∈V∣x∉x}\{ x \in V \mid x \notin x \}, which is the Russell class, resulting in a because this subset would both belong and not belong to itself. Similarly, the axiom of foundation prevents VV from being a set, as it would imply an impossible membership loop like V∈VV \in V. Classes address this limitation by providing a framework for expressing properties that define collections of sets without requiring those collections to be elements of other sets. A class is a definable collection specified by a formula in the language of , allowing one to reason about "all sets" or other large aggregates as classes rather than sets. This enables the treatment of VV as a proper class—a class that is not a set—thus avoiding the need to assume it has set-like membership properties. Proper classes, by definition, cannot be members of any set, which circumvents paradoxes arising from unrestricted comprehension. The introduction of classes offers key benefits for maintaining consistency in set theory. By restricting set membership to "small" collections—those that can be elements—while permitting "large" collections as proper classes, the theory avoids contradictions without overly limiting expressive power. This distinction supports the cumulative hierarchy of sets, where each level builds upon previous ones, but without a total set encompassing the entire . Philosophically, it formalizes the that not all conceivable collections are sets; only those of manageable size qualify, preserving the foundational consistency of .

Definition and Distinction from Sets

Formal Definition of Classes

In , particularly in extensions of with the (ZFC) that incorporate classes, such as von Neumann–Bernays–Gödel (NBG) set theory, a class is formally a collection of sets defined by a first-order ϕ(x)\phi(x) in the of . The elements of the class are precisely those sets xx that satisfy ϕ(x)\phi(x), and the class is denoted by the comprehension notation {x∣ϕ(x)}\{ x \mid \phi(x) \}. This definition arises from the class comprehension axiom schema in NBG, which guarantees the existence of a unique class for any such ϕ(x)\phi(x), where ϕ\phi is predicative (quantifying only over sets) and may include free class variables as parameters. Every set is a class: for any set ss, the ϕ(x)≡x=s\phi(x) \equiv x = s defines a class containing ss as its only element, and by the , sets coincide with their defining classes. However, not every class is a set, as certain classes exceed the size limitations imposed on sets by the axioms of ZFC. An illustrative schema for a class CC in the universe VV of all sets is given by C={x∈V∣ϕ(x)},C = \{ x \in V \mid \phi(x) \}, where VV represents the proper class of all sets, emphasizing that classes partition or select from the entire set-theoretic .

Key Differences Between Sets and Classes

In set theory frameworks such as von Neumann–Bernays–Gödel (NBG), sets and classes serve distinct roles to maintain foundational consistency. Sets are the primary objects that can be elements of other sets or classes, adhering to the , which equates two sets if they share identical elements. They are inherently "small," with their formation constrained by existence axioms like replacement, which ensures that the image of a set under a definable function remains a set, and separation, limiting subclasses to sets when derived from existing sets. In contrast, proper classes cannot be elements of any collection, distinguishing them as "large" totalities that transcend these set-bounding axioms, such as the class Ord comprising all ordinal numbers. Axiomatically, sets operate under the core principles of with choice (ZFC), including axioms of , union, and that guarantee their structured existence. Classes, however, are governed by an extended comprehension schema in NBG, permitting the definition of a class via any φ(x) with parameters (sets or classes) as {x | φ(x)}, without requiring that this class exist as a set. This schema allows classes to capture comprehensive totalities while avoiding the assertion of their set membership, as classes are not subject to ZFC's existence postulates. The structural divergence yields critical implications for paradox avoidance. Since proper classes elude membership, no class of all classes can be formed, circumventing universal that arise from self-referential totalities. Additionally, in ZFC augmented with classes (as in NBG), the power set axiom is restricted to sets alone, asserting that for any set a, the collection of all subsets of a is itself a set, but denying such a construction for proper classes to preserve size limitations. This delineation ensures that classes, definable via formulas as noted earlier, facilitate expressive power without undermining the hierarchy of sets.

Properties and Operations

Basic Properties of Classes

Classes in set theory satisfy the principle of , which states that two classes are equal they have the same elements. This property holds regardless of whether the classes are sets or proper classes, ensuring that classes are determined solely by their membership. Every class is well-defined in the sense that it corresponds to a unique extension given by a defining in the language of , where the formula specifies the precise condition for membership. This ensures that classes are uniquely determined by their extensions, as per extensionality, even if multiple formulas define the same extension. The membership relation ∈\in for classes is restricted such that only sets can be elements of a class; proper classes cannot be members of any class. This distinction maintains the in axiomatic systems like NBG , where classes serve as collections of sets but not of other classes. The subclass relation ⊆\subseteq between classes CC and DD is defined such that C⊆DC \subseteq D every element of CC is also an element of DD. This inclusion relation does not imply equality unless C=DC = D, and it applies uniformly to both sets and proper classes as subclasses. Among the basic classes are the empty class ∅\emptyset, which contains no elements and is itself a set, and the universal class VV, which comprises all sets and is a proper class. These classes exist in NBG set theory by virtue of the axioms of empty set and the class existence theorem, respectively.

Class Operations and Comprehension

In set theory, classes are formed through the comprehension schema, which asserts that for any formula ϕ(x)\phi(x) in the language of set theory, there exists a class C={x∣ϕ(x)}C = \{x \mid \phi(x)\} comprising all elements satisfying ϕ\phi. This schema allows the definition of classes via arbitrary properties, but unlike sets, such a class may be proper if it is too large to be a member of any other class. Basic operations on classes extend those for sets, preserving their definable nature. The union of a class CC, denoted ⋃C\bigcup C, is the class {y∣∃x∈C(y∈x)}\{y \mid \exists x \in C (y \in x)\}, collecting all elements belonging to at least one member of CC. The intersection of two classes AA and BB, denoted A∩BA \cap B, is {x∣x∈A∧x∈B}\{x \mid x \in A \land x \in B\}, consisting of elements common to both. The complement of a class AA relative to the universe VV (the class of all sets) is V∖A={x∈V∣x∉A}V \setminus A = \{x \in V \mid x \notin A\}, though complements are typically defined within restricted contexts to maintain definability. The power class of a class CC, denoted P(C)\mathcal{P}(C), is the class {x∣x⊆C}\{x \mid x \subseteq C\} of all subclasses of CC; if CC is proper, so is P(C)\mathcal{P}(C). Replacement for classes extends the set-theoretic replacement principle: if a ϕ(x,y)\phi(x, y) defines a functional relation (i.e., ∀x∃!y ϕ(x,y)\forall x \exists! y \, \phi(x, y)), then for any class AA, the class {y∣∃x∈A ϕ(x,y)}\{y \mid \exists x \in A \, \phi(x, y)\} exists as the image of AA under this function. However, proper classes impose limitations on operations, as there is no general ensuring that the union or power class of a proper class is itself a set; such constructions yield proper classes, preventing their treatment as elements in further set formations.

Examples of Classes

Definite Descriptions and Small Classes

In , definite descriptions provide a means to specify particular classes through s that uniquely identify their members, often resulting in small classes that coincide with sets. A definite description typically takes the form "the unique xx such that ϕ(x)\phi(x)", where ϕ\phi is a ensuring and ; when such a description defines a collection that is itself an element of another class, it qualifies as a small class, synonymous with a set. For instance, the ∅\emptyset can be described as the unique set containing no elements, formalized as ιx:∀a(a∉x)\iota x : \forall a (a \notin x), and this class is a set by the . A prominent example is the class of natural numbers, denoted ω\omega, which serves as a set-class defined as the smallest infinite ordinal or the intersection of all inductive sets (sets containing ∅\emptyset and closed under the successor operation). This definite description yields ω={x∣x is a finite ordinal}\omega = \{ x \mid x \text{ is a finite ordinal} \}, a countable set that is a member of the universe of sets. Similarly, successor ordinals, such as ω+1\omega + 1, are small classes defined descriptively as the unique ordinal immediately following ω\omega, and they are sets under the axioms of infinity and ordinals. Small classes encompass those collections that are sets, meaning they can be elements of other classes, in contrast to proper classes; this distinction arises from the comprehension axiom, which allows formation of classes via formulas but restricts sets to those that belong to some class. Finite classes, for example, are simply finite sets, such as {∅,{∅}}\{ \emptyset, \{ \emptyset \} \}, which are small by virtue of their bounded size and membership in larger sets like the power set of ω\omega. A key example of a small class is the set of hereditarily finite sets, denoted HF\mathrm{HF} or VωV_\omega, defined as the union of all finite-rank levels of the von Neumann hierarchy: V0=∅V_0 = \emptyset, Vn+1=P(Vn)V_{n+1} = \mathcal{P}(V_n) for finite nn, and Vω=⋃n<ωVnV_\omega = \bigcup_{n < \omega} V_n. This class consists of all sets whose is finite and is itself a set, as established by the axioms of power set and union. By the axioms of , such as separation and replacement, all small classes—those definable and provably elements of some class—are sets, ensuring they remain manageable within the without leading to paradoxes.

Proper Classes and Universal Examples

Proper classes are collections in that are not sets, meaning they cannot be elements of any class and are deemed "too large" to satisfy the axioms defining sets. In axiomatic systems like von Neumann–Bernays–Gödel (NBG) , proper classes arise naturally as unbounded collections that extend beyond the of sets, such as the cumulative hierarchy VV. These classes are essential for describing global structures in the set-theoretic universe without leading to paradoxes, as they are not subject to unrestricted comprehension. The universal class VV, defined by the formula V={x∣x=x}V = \{ x \mid x = x \}, exemplifies a proper class as it encompasses the totality of all sets and cannot itself be a set. If VV were a set, it would lead to inconsistencies akin to , since no set can contain all sets as elements. In NBG, VV serves as a foundational structure, modeling the entire of for . Another prominent proper class is OnOn, the class of all ordinal numbers, which cannot be injected into any set because assuming it were a set would imply the existence of a largest ordinal, contradicting the well-ordered nature of ordinals. Ordinals index the stages of the cumulative hierarchy, and OnOn captures the entire transfinite ordering without bound. Similarly, the class of all cardinals, Card\mathrm{Card}, is proper and well-ordered under the usual ordering of cardinals, as there is no largest cardinal; if Card\mathrm{Card} were a set, it would bound the sizes of sets, which is impossible. The class of all singletons, Sing={{x}∣x∈V}\mathrm{Sing} = \{ \{x\} \mid x \in V \}, provides a further example of a proper class, as it stands in bijective correspondence with VV via the mapping {x}↦x\{x\} \mapsto x. This implies that if Sing\mathrm{Sing} were a set, then VV could be bijected onto a set, which is impossible since proper classes exceed the size of any set. Proper classes like these lack a in the traditional sense applicable to sets but can be compared in size through the existence of class or injections between them.

Paradoxes Involving Classes

Russell's Paradox and Class Comprehension

Russell's paradox emerges from the naive assumption that any definable collection can form a set, specifically when considering the collection RR of all sets that do not contain themselves as members, formally R={x∣x∉x}R = \{ x \mid x \notin x \}. Assuming RR itself is a set leads to a contradiction upon examining its membership: if R∈RR \in R, then by the defining property, R∉RR \notin R; conversely, if R∉RR \notin R, then R∈RR \in R. This yields the logical inconsistency (R∈R)↔¬(R∈R)(R \in R) \leftrightarrow \neg (R \in R), showing that no such set can exist. Bertrand Russell discovered this paradox around 1901, communicated it privately to in 1902, and published it in 1903 in his book The Principles of Mathematics, where it appears in Appendix B as a critical challenge to the foundations of . The paradox reveals the failure of unrestricted comprehension for sets, which permits forming a set from any property without limitations, resulting in self-referential contradictions. In response, the of classes resolves the issue by distinguishing sets from proper classes: sets are collections that can serve as elements of other collections, whereas proper classes, like RR, are too "large" to be sets and thus cannot be members of any class, eliminating the problematic self-membership query. Under this framework, unrestricted comprehension holds for classes but is stratified by size—those classes small enough to be elements qualify as sets, while others remain proper classes, avoiding paradoxes while preserving definability. This class/set distinction, directly motivated by , became a cornerstone of axiomatic set theories developed in the ensuing decades. The , first identified by Cesare Burali-Forti in 1897, emerges from the consideration of the class of all ordinal numbers, denoted On\mathbf{On}. This class appears to be well-ordered and thus an ordinal itself. However, if On\mathbf{On} were a set, its order type would be some ordinal α\alpha, but then α∉On\alpha \notin \mathbf{On} while On\mathbf{On} supposedly contains all ordinals, yielding a contradiction. The paradox is resolved by recognizing On\mathbf{On} as a proper class rather than a set, ensuring that ordinals—defined as sets—cannot encompass all such structures. Cantor's paradox, noted by between 1895 and 1899 in correspondence, involves the power class of the universe VV, denoted P(V)\mathcal{P}(V), which collects all subsets of VV. Cantor's theorem implies no injection exists from P(V)\mathcal{P}(V) into VV, yet assuming a VV exists (containing all sets) leads to P(V)⊆V\mathcal{P}(V) \subseteq V, implying such an injection and thus a contradiction. The resolution treats P(V)\mathcal{P}(V) as a proper class, avoiding the assumption that it forms a set while permitting the collection as a totality in class theory. Grelling's paradox, formulated by Kurt Grelling and Leonard Nelson in 1908, concerns the class of heterological classes—those predicates that do not apply to themselves. The predicate "heterological" leads to a self-referential contradiction: if it is heterological, it applies to itself (contradicting the definition); if not, it fails to apply when it should. Analogous to but framed in terms of predicates, it is addressed by viewing the totality of such classes as a proper class, preventing membership issues inherent to sets. These paradoxes collectively underscore the necessity of proper classes in to accommodate unbounded totalities, such as all ordinals, all power sets, or all self-applicable predicates, without forcing them into set membership and thereby evading contradictory assumptions.

Classes in Axiomatic Set Theories

Von Neumann–Bernays–Gödel (NBG) Set Theory

The Von Neumann–Bernays–Gödel set theory (NBG) emerged as a foundational in the early to address limitations in treating large collections within standard set theories. laid the groundwork in 1925 by axiomatizing with classes as primitive entities, defined via functions indicating membership, to avoid unrestricted comprehension while enabling broader mathematical constructions. Paul Bernays advanced this in 1937 through a series of papers, introducing a clearer distinction between sets (which can be elements) and proper classes (which cannot), and refining the axioms to ensure consistency with Zermelo-Fraenkel principles. formalized and simplified the system in 1940, using it as the basis for proving the relative consistency of the and the generalized ; his version emphasized and to prevent paradoxes like Russell's. NBG's axioms build directly on those of Zermelo–Fraenkel set theory with (ZFC) but extend them to include classes explicitly. The core ZFC axioms—, or pairing, union, , , regularity (foundation), replacement schema, and —are restricted to sets, ensuring that sets behave as in ZFC. Additional axioms govern classes: for classes (two classes are equal if they have the same set members), and an existence axiom asserting the universal class VV of all sets. The theory's primitive relation is membership ∈\in, applying between sets and classes or sets and sets, but proper classes cannot be members to avoid vicious circles. Central to NBG is the class comprehension schema, which permits the formation of any class definable by a formula without unbound class quantifiers. Formally, for any formula ϕ(x)\phi(x) in the language of first-order set theory (with parameters from sets or classes, but quantifiers ranging only over sets), there exists a unique class CC such that ∀x(x∈C↔ϕ(x)),\forall x (x \in C \leftrightarrow \phi(x)), where xx ranges over sets. This schema replaces ZFC's separation and replacement schemas for sets, deriving them as special cases, and allows classes like the class of all ordinals or the class of all singletons, which are proper classes in ZFC. It resolves paradoxes by treating proper classes as non-elements, enabling their use in definitions and proofs without collapsing the hierarchy. NBG includes a global choice axiom, extending the axiom of choice beyond sets to the entire universe of classes. This asserts the existence of a class FF that is a choice function on all nonempty classes of disjoint nonempty sets, or equivalently, a class well-ordering of the universe VV. In practice, it guarantees a selector for every nonempty class of pairwise disjoint nonempty sets, facilitating arguments involving proper classes, such as in model constructions. Unlike ZFC's set choice, global choice applies universally but follows from the in Gödel's framework. NBG is equiconsistent with ZFC, meaning if ZFC is consistent, so is NBG, and vice versa; moreover, NBG is a conservative extension of ZFC, proving exactly the same theorems about sets. This equivalence arises because classes in NBG can be interpreted as definable subsets via s in ZFC models: given a ZFC model MM, extend it to include classes as pairs (ϕ,M)(\phi, M) where ϕ\phi is a , preserving satisfaction. Gödel established this in his 1940 work, showing no new set-theoretic truths emerge from classes.

Morse–Kelley (MK) Set Theory

Morse–Kelley set theory, often abbreviated as MK or Kelley-Morse set theory, is an axiomatic framework that builds upon the structure of von Neumann–Bernays–Gödel (NBG) set theory by incorporating a more powerful comprehension principle for classes. In this system, sets and classes are distinguished, with sets forming a subclass of classes, and the axioms ensure that every set is a class while proper classes cannot be elements of other classes. The core axioms mirror those of NBG, including extensionality, pairing, union, power set, infinity, replacement (for sets), regularity (foundation), and the axiom of choice, along with schemas for the existence of classes defined by first-order properties of sets. However, the distinguishing feature is the second-order class comprehension schema, which permits the definition of classes using formulas that include quantifiers over arbitrary classes, not just sets. This second-order comprehension enables impredicative definitions, allowing a class to be specified by a property that quantifies over the entire collection of classes, including the class being defined itself. For example, the class of all sets that are not elements of themselves can be impredicatively comprehended in a way that captures global properties across the . Such impredicativity provides MK with greater expressive power for describing hierarchies beyond what first-order comprehension in NBG allows, facilitating the formalization of advanced concepts in and logic that require referencing the full universe of classes. The development of MK traces back to foundational work by Anthony P. Morse in the 1940s on impredicative class theories, which was later formalized and popularized by John L. Kelley in his 1955 textbook General Topology, where the axioms were presented as a rigorous extension of Zermelo-Fraenkel set theory with classes. MK interprets NBG faithfully, meaning that theorems of NBG about sets are provable in MK, but the additional strength of second-order comprehension equips MK to handle taller cumulative hierarchies and more complex definability notions. Regarding consistency strength, MK is equiconsistent with ZFC plus the existence of a strongly inaccessible cardinal: if ZFC proves the existence of an inaccessible cardinal κ, then the structure ⟨V_κ, ∈, P(V_κ)⟩ models MK, and conversely, the consistency of MK implies the consistency of ZFC + "there exists an inaccessible cardinal." Furthermore, MK + V = L is inconsistent, as MK's consistency strength exceeds that of ZFC + V = L, which proves there are no inaccessible cardinals.

Other Formal Systems

Beyond the standard axiomatic set theories like NBG and MK, several alternative formal systems incorporate classes in distinct ways, often to address specific foundational concerns such as the inclusion of urelements or restrictions on comprehension principles. These systems provide frameworks for handling proper classes while exploring variations in consistency strength and expressive power. One extension involves ZFA set theory, which modifies ZF by permitting urelements (atoms) as primitive non-set objects that can be members of sets but have no elements themselves. In such systems, classes are treated analogously to those in ZF-based theories, allowing comprehension over formulas that may involve atoms, thereby enabling models where urelements form proper classes while preserving much of the structure of standard set theory. This approach facilitates the study of permutation models and symmetry in set-theoretic constructions, as detailed in foundational treatments of urelement theories. Kripke-Platek set theory (KP) can be augmented with classes to focus on admissible structures, particularly those aligned with admissible ordinals, where the universe is modeled as a up to such ordinals. Here, class comprehension is restricted to recursive definitions, typically Δ₀-formulas or those definable within the admissible set, ensuring that classes remain "tame" and computably describable. This formulation supports recursion theory and definability in admissible contexts, as explored in seminal works on admissible sets. Non-standard theories like NFU (New Foundations with urelements), introduced by Jensen in 1969, employ stratified comprehension to define classes while allowing urelements, thereby avoiding paradoxes through type-theoretic restrictions on formulas. In NFU, all entities are classes (including sets as membered classes), and is weakened to accommodate atoms, enabling the construction of large stratified collections without the full power set axiom. Jensen proved NFU consistent relative to a weak subsystem of ZFC, specifically interpretable in . Category-theoretic approaches reframe classes as objects within a , such as in topos theory, where the topos itself acts as a generalized of "sets" and "classes" via internal logic and sheaf semantics. In this setting, proper classes correspond to large objects or global sections that transcend small set-like subcategories, providing a structural alternative to axiomatic hierarchies while preserving set-theoretic intuitions through categorical universals. These systems exhibit varying consistency strengths; for instance, NFU is consistent relative to assumptions weaker than full ZFC, such as Zermelo set theory without replacement, highlighting its relative equiconsistency with modest arithmetic subsystems compared to the stronger implications of ZFC.

Advanced Topics

Global Choice and Classes

The axiom of global choice is a strengthening of the axiom of choice that applies to the entire universe of sets in theories incorporating proper classes, such as von Neumann–Bernays–Gödel (NBG) set theory. It asserts the existence of a single class CC that functions as a choice function for all non-empty sets simultaneously, meaning CC is a class such that for every non-empty set SS, there is a unique y∈Sy \in S with y=C(S)y = C(S). This extends the selection principle beyond finite or countable collections to the full hierarchy of sets, ensuring a uniform mechanism for picking elements across the entire cumulative hierarchy VV. The axiom of global choice is equivalent to the existence of a well-ordering of the as a class. While it implies and is implied by the standard for sets, it remains independent of the core axioms of NBG, as demonstrated by models where choice holds for sets but no global selector exists. This independence highlights its status as an additional assumption, often denoted as GBC when added to the base theory. The axiom implies the standard but goes further by enabling coordinated selections that transcend set-sized families. A primary implication of global choice is its capacity to well-order proper classes, including the class of all sets VV and the class of ordinals On\mathrm{On}, thereby providing a linear ordering for structures that are too large to be sets. This facilitates proofs involving transfinite and ordinal indexing across the entire . Notably, the axiom is equivalent to the existence of a well-ordering of the as a class, meaning there is a class relation that well-orders all sets in a manner compatible with the membership .

Limitations and Open Questions

One key limitation in the theory of classes arises from the absence of a "class of all classes," which parallels the paradoxes encountered with sets but introduces meta-theoretic challenges. In systems like NBG and MK, while the universal class VV encompasses all sets, the collection of all classes cannot itself be formalized as a class without risking inconsistency, as classes are definable predicates that would require quantifying over themselves in an unrestricted manner. This restriction ensures avoidance but leaves open questions about the structure of the "class universe" beyond VV, preventing a fully closed of collections. A prominent open question concerns the precise consistency strength of second-order class theories such as MK relative to axioms. While MK is known to exceed the strength of ZFC plus the existence of a transitive model of ZFC, its exact position in the hierarchy of remains undetermined, as it does not align neatly with standard notions like inaccessibles or Mahlos. This gap highlights the challenge of calibrating class theories against the well-ordered consistency scale provided by in first-order . The impredicativity inherent in class comprehension schemas, particularly in MK where formulas may quantify over all classes, can lead to unintended proof-theoretic strengths. For instance, MK proves statements beyond what its intended models might support, such as full over the universe, potentially overreaching the bounds of predicative constructions. This feature enhances expressive power but raises concerns about the theory's alignment with iterative conceptions of sets and classes. Philosophically, whether the universe VV qualifies as a class within some higher-order persists as an unresolved issue, blending foundational and metaphysical considerations without a definitive mathematical resolution. Set theorists debate this in terms of potentialism versus , where VV might be viewed as a proper class in an extended framework, but no consensus exists on its ontological status. Recent developments have explored connections between classes in set theory and universes in (HoTT), where cumulative universes serve roles analogous to proper classes by classifying types without forming paradoxical totalities. Post-2010s work in univalent foundations models set-theoretic classes via these universes, offering a synthetic alternative that avoids some classical limitations while preserving expressive depth.

References

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