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Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω2

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, a successor ordinal, or a limit ordinal.

For example, the smallest limit ordinal is ω, the smallest ordinal greater than every natural number. This is a limit ordinal because for any smaller ordinal (i.e., for any natural number) n we can find another natural number larger than it (e.g. n+1), but still less than ω. The next-smallest limit ordinal is ω+ω. This will be discussed further in the article.

Using the von Neumann definition of ordinals, every ordinal is the well-ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal.

Alternative definitions

[edit]

Various other ways to define limit ordinals are:

  • It is equal to the supremum of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)
  • It is not zero and has no maximum element.
  • It can be written in the form ωα for α > 0. That is, in the Cantor normal form there is no finite number as last term, and the ordinal is nonzero.
  • It is a limit point of the class of ordinal numbers, with respect to the order topology. (The other ordinals are isolated points.)

Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor; some textbooks include 0 in the class of limit ordinals[1] while others exclude it.[2]

Examples

[edit]

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n. Taking the union (the supremum operation on any set of ordinals) of all the ω·n, we get ω·ω = ω2, which generalizes to ωn for any natural number n. This process can be further iterated as follows to produce:

In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still countable ordinals. However, there is no recursively enumerable scheme for systematically naming all ordinals less than the Church–Kleene ordinal, which is a countable ordinal.

Beyond the countable, the first uncountable ordinal is usually denoted ω1. It is also a limit ordinal.

Continuing, one can obtain the following (all of which are now increasing in cardinality):

In general, we always get a limit ordinal when taking the union of a nonempty set of ordinals that has no maximum element.

The ordinals of the form ω²α, for α > 0, are limits of limits, etc.

Properties

[edit]

The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable.

If we use the von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point): the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.

Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).

Indecomposable ordinals

[edit]
Main article: Indecomposable ordinal

Additively indecomposable

A limit ordinal α is called additively indecomposable if it cannot be expressed as the sum of β < α ordinals less than α. These numbers are any ordinal of the form for β an ordinal. The smallest is written , the second is written , etc.[3]

Multiplicatively indecomposable

A limit ordinal α is called multiplicatively indecomposable if it cannot be expressed as the product of β < α ordinals less than α. These numbers are any ordinal of the form for β an ordinal. The smallest is written , the second is written , etc.[3]

Exponentially indecomposable and beyond

The term "exponentially indecomposable" does not refer to ordinals not expressible as the exponential product (?) of β < α ordinals less than α, but rather the epsilon numbers, "tetrationally indecomposable" refers to the zeta numbers, "pentationally indecomposable" refers to the eta numbers, etc.[3]

See also

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References

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Further reading

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Limit ordinal

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In set theory, a limit ordinal is an ordinal number that is neither zero nor the successor of any other ordinal, meaning it lacks an immediate predecessor and instead serves as the least upper bound (supremum) of all smaller ordinals. This distinguishes limit ordinals from successor ordinals, which are of the form α+1=α∪{α}\alpha + 1 = \alpha \cup \{\alpha\} for some ordinal α\alpha. The smallest limit ordinal is ω\omega, which is the order type of the natural numbers and the first infinite ordinal. Other examples include ω⋅2=sup⁡{ω⋅1+n∣n<ω}\omega \cdot 2 = \sup\{\omega \cdot 1 + n \mid n < \omega\}, ω2\omega^2, and more generally, any ordinal of the form ωα\omega^\alpha for a limit ordinal α\alpha. Limit ordinals play a crucial role in the transfinite hierarchy, as every nonzero ordinal is either a successor or a limit, and they mark points where the ordinal arithmetic transitions from finite-like addition to suprema of sequences. A key example of a larger limit ordinal is ω1\omega_1, the least uncountable ordinal, which is the supremum of all countable ordinals and serves as the initial ordinal of cardinality ℵ1\aleph_1. Limit ordinals are characterized by their cofinality, the smallest cardinality of a cofinal subset, with regular limit ordinals (like ω\omega and ω1\omega_1 under the axiom of choice) having cofinality equal to themselves. In the von Neumann construction, a limit ordinal λ\lambda equals the union ⋃γ∈λγ\bigcup_{\gamma \in \lambda} \gamma, emphasizing its role as a "limit point" in the class of all ordinals.

Definition and Basics

Formal Definition

In set theory, ordinal numbers are formally defined using the von Neumann construction, where an ordinal α\alpha is a transitive set that is well-ordered by the membership relation ∈\in. Specifically, α\alpha is transitive if every element of α\alpha is a subset of α\alpha, and it is well-ordered by ∈\in if every nonempty subset of α\alpha has a least element with respect to ∈\in. Under this definition, the order on ordinals is given by β<α\beta < \alpha if and only if β∈α\beta \in \alpha, and each ordinal α\alpha consists precisely of all ordinals strictly smaller than itself, i.e., α={β∣β<α}\alpha = \{\beta \mid \beta < \alpha\}. A limit ordinal λ\lambda is an ordinal that is neither zero nor a successor ordinal. A successor ordinal is one of the form α+1\alpha + 1 for some ordinal α\alpha, defined as α+1=α∪{α}\alpha + 1 = \alpha \cup \{\alpha\}. Thus, λ\lambda has no immediate predecessor, meaning there is no ordinal α\alpha such that λ=α+1\lambda = \alpha + 1. Equivalently, every limit ordinal λ\lambda satisfies λ=sup⁡{β∣β<λ}=⋃λ\lambda = \sup \{\beta \mid \beta < \lambda\} = \bigcup \lambda, the least upper bound (supremum) of the set of all ordinals strictly less than λ\lambda. This supremum characterization implies that the set {β∣β<λ}\{\beta \mid \beta < \lambda\} is nonempty (for λ>0\lambda > 0) and has no maximum element, so there is no largest ordinal less than λ\lambda. The zero ordinal, defined as the empty set ∅\emptyset, is often classified as a limit ordinal because it is not a successor and satisfies sup⁡∅=0\sup \emptyset = 0, though it is frequently treated separately in discussions due to its unique position as the smallest ordinal with no elements at all.

Relation to Successor Ordinals

A successor ordinal σ\sigma is defined as σ=α+1\sigma = \alpha + 1 for some ordinal α\alpha, meaning it immediately follows α\alpha in the ordinal ordering and possesses a direct predecessor. In the von Neumann representation of ordinals as sets, a successor ordinal includes α\alpha as its maximum element, ensuring a discrete "step" in the well-ordered structure. In contrast, limit ordinals lack such an immediate predecessor, positioning them as points of accumulation or transition in the ordinal hierarchy where no single ordinal suffices as the largest below them. This absence of a maximum element below a limit ordinal creates conceptual "gaps" in the of ordinals, distinguishing them from the isolated steps of successor ordinals. The class of all non-zero ordinals partitions exhaustively into successor ordinals and limit ordinals, with no overlap, reflecting the foundational dichotomy in transfinite arithmetic. Within transfinite sequences or constructions, limit ordinals emerge precisely at limit stages, where the sequence approaches without a final immediate prior term, underscoring their role in capturing infinite progressions.

Characterizations

Topological View

The order topology on a set of ordinals is generated by taking as a subbasis the collection of all open rays of the form (α,∞)={β∣α<β}(\alpha, \infty) = \{\beta \mid \alpha < \beta\} and (−∞,γ)={β∣β<γ}(-\infty, \gamma) = \{\beta \mid \beta < \gamma\} for ordinals α,γ\alpha, \gamma, or equivalently, by using open intervals (α,β)={γ∣α<γ<β}(\alpha, \beta) = \{\gamma \mid \alpha < \gamma < \beta\} as a basis. This topology reflects the linear order structure of the ordinals, making the space Hausdorff and rendering initial segments [0,α)[0, \alpha) compact if and only if α\alpha is a successor ordinal. In this topology, successor ordinals are isolated points. For a successor ordinal σ=ρ+1\sigma = \rho + 1, the singleton {σ}\{\sigma\} forms an open set, as it equals the interval (ρ,σ+1)(\rho, \sigma + 1), separating σ\sigma from all other ordinals. In contrast, limit ordinals λ\lambda are limit points: every open neighborhood of λ\lambda, such as (α,β)(\alpha, \beta) with α<λ<β\alpha < \lambda < \beta, contains ordinals both strictly less than λ\lambda and strictly greater than λ\lambda, with no immediate predecessor to λ\lambda itself. More precisely, λ\lambda serves as a limit point of the set {β∣β<λ}\{\beta \mid \beta < \lambda\}, since any neighborhood of λ\lambda intersects this set in infinitely many points due to the well-ordering. This topological perspective extends naturally to the proper class Ord\mathrm{Ord} of all ordinals, endowed with the order topology generated similarly by open intervals and rays across the entire class. In Ord\mathrm{Ord}, limit ordinals continue to act as limit points, accumulating the ordinals below them, while successor ordinals remain isolated, highlighting the "discrete" nature of successors amid the continuous accumulation at limits. This structure underscores the topological distinction between the discrete steps of successor ordinals and the convergence properties at limit ordinals.

Normal Form Representation

Every ordinal can be uniquely expressed in Cantor normal form as a finite sum of the form α=ωβ1⋅k1+ωβ2⋅k2+⋯+ωβn⋅kn\alpha = \omega^{\beta_1} \cdot k_1 + \omega^{\beta_2} \cdot k_2 + \cdots + \omega^{\beta_n} \cdot k_n, where β1>β2>⋯>βn≥0\beta_1 > \beta_2 > \cdots > \beta_n \geq 0 are ordinals and each kik_i is a positive finite .\] [](https://people.maths.ox.ac.uk/knight/lectures/moreordinals.pdf) This representation provides an algebraic structure analogous to polynomial expansion in base $\omega$, facilitating computations in [ordinal arithmetic](/page/Ordinal_arithmetic).\[ A nonzero ordinal λ\lambda is a limit ordinal if and only if its Cantor normal form contains no term with exponent , meaning there is no finite addend ω0⋅k\omega^0 \cdot k (with k≥1k \geq 1) at the end of the expansion.\] [](https://people.maths.ox.ac.uk/knight/lectures/moreordinals.pdf) Equivalently, such limit ordinals can be expressed as $\lambda = \omega \cdot \mu$ for some ordinal $\mu \geq 1$, where the multiplication shifts the exponents in the normal form of $\mu$ upward by 1, ensuring no [constant term](/page/Constant_term) remains.\[ This absence of a finite term reflects the defining property of limit ordinals: they have no immediate predecessor and are the supremum of all smaller ordinals.[]

Examples

Countable Examples

The smallest countable limit ordinal is ω\omega, which is the of the natural numbers and the supremum of all finite ordinals, sup⁡{n∣n<ω}\sup\{n \mid n < \omega\}. This ordinal represents the first infinite point in the hierarchy of ordinals, with no immediate predecessor, as every ordinal less than ω\omega is finite. A simple extension is ω⋅2=ω+ω\omega \cdot 2 = \omega + \omega, the obtained by concatenating two copies of the natural numbers, which is the supremum sup⁡{ω+n∣n<ω}\sup\{\omega + n \mid n < \omega\}. This illustrates how limit ordinals can be built by repeating the structure of ω\omega a finite number of times and then taking a limit. More generally, for each finite n≥1n \geq 1, ω⋅n\omega \cdot n is a countable limit ordinal formed by nn copies of ω\omega, and the supremum of these, sup⁡{ω⋅n∣n<ω}=ω2\sup\{\omega \cdot n \mid n < \omega\} = \omega^2, represents the order type of ω×ω\omega \times \omega under the lexicographic order. This ω2\omega^2 captures a quadratic growth in the ordinal hierarchy, achievable through countable iterations. Higher finite powers follow similarly: ω3\omega^3, ω4\omega^4, and so on up to ωω=sup⁡{ωn∣n<ω}\omega^\omega = \sup\{\omega^n \mid n < \omega\}, which is the limit of exponentiating ω\omega over all finite exponents and embodies the first countable ordinal closed under finite exponentiation. Finally, ε0\varepsilon_0 is the first fixed point of the function α↦ωα\alpha \mapsto \omega^\alpha, defined as the supremum sup⁡{ω,ωω,ωωω,… }\sup\{\omega, \omega^\omega, \omega^{\omega^\omega}, \dots \}, where the sequence builds taller and taller towers of exponentiation. This ordinal remains countable despite its immense size, marking a significant milestone in the countable ordinal hierarchy used in proof theory.

Uncountable Examples

Uncountable limit ordinals represent a significant escalation in the hierarchy of ordinals, marking the transition from countable to uncountable transfinite structures and serving as foundational elements in advanced set-theoretic constructions such as the and forcing extensions. These ordinals are neither zero nor successors, arising as suprema of unbounded increasing sequences of smaller ordinals, and their uncountable nature introduces cardinalities beyond ℵ0\aleph_0, influencing concepts like stationary sets and reflection principles. The smallest uncountable ordinal, ω1\omega_1, is a canonical example of an uncountable limit ordinal, defined as the least upper bound of the set of all countable ordinals: ω1=sup⁡{α∣α is countable}\omega_1 = \sup\{\alpha \mid \alpha \text{ is countable}\}. This makes ω1\omega_1 the first ordinal whose underlying set has uncountable cardinality, embodying the accumulation of all countable order types without an immediate predecessor. Following ω1\omega_1, the ordinal ω2\omega_2 provides another example, constructed as the supremum of all ordinals α\alpha such that ∣α∣≤ℵ1|\alpha| \leq \aleph_1: ω2=sup⁡{α∣∣α∣≤ℵ1}\omega_2 = \sup\{\alpha \mid |\alpha| \leq \aleph_1\}. More generally, for any ordinal α≥1\alpha \geq 1, the initial ordinals ωα\omega_\alpha are uncountable limit ordinals, forming the backbone of the aleph hierarchy and enabling the enumeration of initial ordinals of successively larger cardinalities. Infinite cardinals offer a broad class of uncountable limit ordinals under the von Neumann construction, where each infinite cardinal κ\kappa is identified with the least ordinal of that cardinality, denoted ord⁡(κ)=κ\operatorname{ord}(\kappa) = \kappa. In this representation, κ=sup⁡{β∣β<κ}\kappa = \sup\{\beta \mid \beta < \kappa\}, ensuring it is a limit ordinal with no largest proper initial segment. Examples include ℵ1=ω1\aleph_1 = \omega_1 and ℵ2=ω2\aleph_2 = \omega_2, but the property extends to all infinite cardinals, such as measurable cardinals, which are uncountable strong limits. The ordinal ωω\omega_\omega, defined as the supremum ωω=sup⁡{ωn∣n<ω}\omega_\omega = \sup\{\omega_n \mid n < \omega\}, exemplifies an uncountable limit ordinal arising from iterating the aleph function over the countable ordinals, resulting in a structure whose cardinality is ℵω\aleph_\omega. Similarly, beth numbers provide further instances: for a limit ordinal λ\lambda, ℶλ=sup⁡{ℶα∣α<λ}\beth_\lambda = \sup\{\beth_\alpha \mid \alpha < \lambda\} is a limit cardinal, hence a limit ordinal. The continuum, the least ordinal of cardinality ℶ1=2ℵ0\beth_1 = 2^{\aleph_0} (the cardinality of the power set of the naturals), is an uncountable limit ordinal, independent of whether the holds, as its value exceeds ω\omega and accumulates all smaller power sets.

Properties

Closure Under Operations

Limit ordinals exhibit specific closure properties under the basic operations of ordinal arithmetic, particularly when at least one operand is a limit ordinal. In ordinal addition, the sum α+λ\alpha + \lambda is a limit ordinal whenever λ\lambda is a limit ordinal, for any ordinal α\alpha. This follows from the definition of addition at limit stages: α+λ=sup⁡{α+β∣β<λ}\alpha + \lambda = \sup\{\alpha + \beta \mid \beta < \lambda\}, where the supremum over the unbounded set of predecessors β<λ\beta < \lambda yields a limit ordinal, as no single predecessor reaches it. Consequently, the sum of two limit ordinals λ+μ\lambda + \mu is itself a limit ordinal provided μ>0\mu > 0. However, limit ordinals are not closed under addition when a positive finite ordinal is added on the right. For a limit ordinal λ\lambda and positive finite n>0n > 0, λ+n\lambda + n is a successor ordinal, specifically (λ+(n−1))+1(\lambda + (n-1)) + 1. This non-closure highlights the non-commutative nature of ordinal addition, though the focus here remains on cases involving limit operands. Under ordinal multiplication, the product α⋅λ\alpha \cdot \lambda is a limit ordinal for any non-zero ordinal α\alpha and limit ordinal λ\lambda. This preservation arises because α⋅λ=⋃β<λ(α⋅β)\alpha \cdot \lambda = \bigcup_{\beta < \lambda} (\alpha \cdot \beta), and the union over an unbounded chain of ordinals below λ\lambda cannot be a successor, as it lacks a maximal element. Thus, the product of two limit ordinals λ⋅μ\lambda \cdot \mu (with λ≠0\lambda \neq 0) is a limit ordinal when μ\mu is limit. For instance, ω+ω=ω⋅2\omega + \omega = \omega \cdot 2 is a limit ordinal. Ordinal exponentiation further demonstrates closure for limit ordinals. Specifically, for any ordinal α>1\alpha > 1 and limit ordinal λ\lambda, αλ\alpha^\lambda is a limit ordinal, defined as αλ=sup⁡{αβ∣β<λ}\alpha^\lambda = \sup\{\alpha^\beta \mid \beta < \lambda\}, which again forms an unbounded supremum without a predecessor. In particular, ωλ\omega^\lambda is a limit ordinal for any limit ordinal λ>0\lambda > 0. Additionally, if λ\lambda is limit and the exponent μ≠0\mu \neq 0, then λμ\lambda^\mu is also a limit ordinal.

Role in Transfinite Induction

Transfinite induction generalizes the principle of mathematical induction to the class of all ordinals, allowing proofs of properties that hold across the entire well-ordered hierarchy of ordinals. To establish that a property P(α)P(\alpha) holds for every ordinal α\alpha, one assumes P(β)P(\beta) for all β<α\beta < \alpha and derives P(α)P(\alpha). For successor ordinals α=γ+1\alpha = \gamma + 1, this typically relies on the immediate predecessor γ\gamma. However, at limit ordinals λ\lambda, which lack an immediate predecessor, the proof of P(λ)P(\lambda) must leverage the uniformity of PP across all β<λ\beta < \lambda, often by showing that P(λ)P(\lambda) follows from the collection of prior instances, such as through a supremum or union operation that captures the "limit" behavior. This structure is essential in transfinite recursion, where functions or sequences are defined hierarchically over ordinals. A recursive definition specifies f(0)f(0) at the base, f(α+1)=g(f(α))f(\alpha + 1) = g(f(\alpha)) at successors using a given operation gg, and at limit ordinals λ\lambda, f(λ)=h({f(β)∣β<λ})f(\lambda) = h(\{f(\beta) \mid \beta < \lambda\}), where hh aggregates previous values—commonly the union for set constructions. This ensures the recursion "converges" at limits by applying continuous operations to the entire preceding stage, preventing gaps in the definition. Such recursions underpin many set-theoretic constructions, maintaining well-definedness across the transfinite. Limit ordinals play a pivotal role as stages for applying continuous operations in advanced hierarchies, exemplified by Gödel's constructible universe LL. Here, the levels LαL_\alpha are built by transfinite recursion: L0=∅L_0 = \emptyset, Lα+1=def⁡(Lα)L_{\alpha+1} = \operatorname{def}(L_\alpha) (the definable subsets of LαL_\alpha), and for limit λ\lambda, Lλ=⋃β<λLβL_\lambda = \bigcup_{\beta < \lambda} L_\beta, incorporating all prior constructible sets without introducing new definitions at the limit itself. This union ensures closure and continuity, forming the foundation for relative consistency proofs in set theory. Historically, Georg Cantor employed limit ordinals to transcend finite counting and successor constructions, introducing them as the suprema of increasing sequences of ordinals to generate higher transfinite numbers and explore infinite hierarchies.

Cofinality

Cofinality Concept

The cofinality of a limit ordinal λ\lambda provides a measure of how λ\lambda can be approached by smaller ordinals, capturing the "density" of sequences leading up to it without a maximum element. It quantifies the minimal complexity required to reach λ\lambda as a supremum through an increasing sequence of proper initial segments. This concept is fundamental in set theory for distinguishing the structural properties of limit ordinals beyond their cardinality. Formally, the cofinality cf⁡(λ)\operatorname{cf}(\lambda) is defined as the smallest ordinal δ\delta such that there exists a strictly increasing cofinal map f:δ→λf: \delta \to \lambda, meaning f(γ)<f(γ′)f(\gamma) < f(\gamma') for all γ<γ′\gamma < \gamma' and sup⁡f′′δ=λ\sup f''\delta = \lambda, where f′′δf''\delta denotes the range of ff. Equivalently, cf⁡(λ)=min⁡{∣C∣∣C⊆λ is cofinal in λ}\operatorname{cf}(\lambda) = \min \{ |C| \mid C \subseteq \lambda \text{ is cofinal in } \lambda \}, where a subset CC is cofinal if for every α<λ\alpha < \lambda, there exists β∈C\beta \in C with α≤β\alpha \leq \beta, and ∣C∣|C| is the cardinality of CC. For any limit ordinal λ\lambda, cf⁡(λ)≤λ\operatorname{cf}(\lambda) \leq \lambda and cf⁡(λ)\operatorname{cf}(\lambda) is always a regular cardinal, meaning cf⁡(cf⁡(λ))=cf⁡(λ)\operatorname{cf}(\operatorname{cf}(\lambda)) = \operatorname{cf}(\lambda). A limit ordinal λ\lambda is regular if cf⁡(λ)=λ\operatorname{cf}(\lambda) = \lambda, as exemplified by ω\omega (the first infinite ordinal) and ω1\omega_1 (the first uncountable ordinal); otherwise, it is singular if cf⁡(λ)<λ\operatorname{cf}(\lambda) < \lambda. Every limit ordinal has cofinality greater than 0, reflecting the absence of a largest element, whereas successor ordinals α+1\alpha + 1 have cf⁡(α+1)=1\operatorname{cf}(\alpha + 1) = 1. This distinction underscores cofinality's role in characterizing the least "step size" needed to cofinally map into the ordinal.

Types of Limit Ordinals by Cofinality

Limit ordinals are classified primarily by the relationship between their cofinality and themselves, leading to regular and singular types. A regular limit ordinal λ\lambda satisfies \cf(λ)=λ\cf(\lambda) = \lambda, meaning it cannot be reached as the supremum of fewer than λ\lambda many smaller ordinals. This property implies that λ\lambda is a regular cardinal. Examples include the smallest infinite ordinal ω\omega, where \cf(ω)=ω\cf(\omega) = \omega, and the first uncountable ordinal ω1\omega_1, with \cf(ω1)=ω1\cf(\omega_1) = \omega_1. Larger instances are inaccessible cardinals, which are uncountable regular strong limit cardinals. In contrast, a singular limit ordinal λ\lambda has \cf(λ)<λ\cf(\lambda) < \lambda, allowing it to be expressed as the supremum of a proper initial segment of smaller ordinals. Such ordinals are not regular cardinals. Representative examples are ωω\omega_\omega, the supremum of the sequence ωn\omega_n for finite nn, with \cf(ωω)=ω\cf(\omega_\omega) = \omega, and the cardinal ℵω\aleph_\omega, defined as sup⁡{ℵn∣n<ω}\sup\{\aleph_n \mid n < \omega\}, also satisfying \cf(ℵω)=ω\cf(\aleph_\omega) = \omega. Among limit cardinals, which are infinite cardinals that are not successor cardinals (i.e., of the form ℵδ\aleph_\delta for limit ordinals δ\delta), a subclass consists of those with uncountable cofinality. These include regular limit cardinals like ω1\omega_1 and certain singular limit cardinals, such as ℵω1\aleph_{\omega_1} where \cf(ℵω1)=ω1>ω\cf(\aleph_{\omega_1}) = \omega_1 > \omega. Strong limit cardinals form another significant class, defined such that 2μ<κ2^\mu < \kappa for all cardinals μ<κ\mu < \kappa; these are necessarily limit cardinals and frequently possess high cofinality, including uncountable or even regular cofinality in prominent cases like inaccessible cardinals. This classification by cofinality has structural implications in set theory: singular limit ordinals enable compression in transfinite hierarchies by allowing larger ordinals to be generated from sequences of smaller cofinality, facilitating more efficient constructions in ordinal arithmetic and large cardinal hierarchies.

Special Classes

Additively Indecomposable Ordinals

An additively indecomposable ordinal γ>0\gamma > 0 is defined as one satisfying the condition that for all ordinals α,β<γ\alpha, \beta < \gamma, α+β<γ\alpha + \beta < \gamma. This property implies that γ\gamma cannot be expressed as the sum of two non-zero ordinals both strictly smaller than γ\gamma. By , the additively indecomposable ordinals are precisely those of the form ωδ\omega^\delta for some ordinal δ\delta. Representative examples include ω=ω1\omega = \omega^1, ωω\omega^\omega, and ε0\varepsilon_0, the least ordinal satisfying ωε0=ε0\omega^{\varepsilon_0} = \varepsilon_0. Further examples arise in the Veblen hierarchy, where the functions φα\varphi_\alpha enumerate classes of such ordinals, with φ0(β)=ωβ\varphi_0(\beta) = \omega^\beta generating the base level of additively indecomposables and higher levels producing fixed points that remain of this form. In Cantor normal form, every additively indecomposable ordinal γ=ωδ\gamma = \omega^\delta is expressed as a single term ωδ⋅1\omega^\delta \cdot 1, with no additional summands. For distinct δ,ε\delta, \varepsilon, ωδ+ωε=ωmax⁡(δ,ε)\omega^\delta + \omega^\varepsilon = \omega^{\max(\delta, \varepsilon)}, which is again additively indecomposable.

Multiplicatively Indecomposable Ordinals

A limit ordinal γ\gamma is multiplicatively indecomposable if, whenever γ=α⋅β\gamma = \alpha \cdot \beta for ordinals α\alpha and β\beta, then α=0\alpha = 0 or β=0\beta = 0 or β=1\beta = 1. This condition ensures that γ\gamma cannot be expressed as a non-trivial product involving a right factor greater than 1. Equivalently, for limit ordinals greater than 1, no such γ\gamma can be written as the product of two ordinals both strictly smaller than γ\gamma and greater than 1. The infinite multiplicatively indecomposable ordinals take the form ωωδ\omega^{\omega^\delta} for some ordinal δ≥0\delta \geq 0. This structure arises because ordinal multiplication corresponds to addition in the exponents: if α=ωμ\alpha = \omega^\mu and β=ων\beta = \omega^\nu, then α⋅β=ωμ+ν\alpha \cdot \beta = \omega^{\mu + \nu}. For the product to remain below ωωδ\omega^{\omega^\delta}, the exponent ωδ\omega^\delta must itself be additively indecomposable, ensuring that sums of smaller exponents stay strictly below it. In the Veblen hierarchy, higher multiplicatively indecomposable ordinals appear as fixed points beyond the epsilon numbers, such as those enumerated by functions like φ(ωδ,0)\varphi(\omega^\delta, 0). Representative examples include ωω\omega^\omega, the least infinite multiplicatively indecomposable ordinal greater than ω\omega, obtained as the supremum of {ωn∣n<ω}\{\omega^n \mid n < \omega\}. The next is ωωω\omega^{\omega^\omega}, and continuing this process yields the tower \omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}\) with nlevelsforeachfinitelevels for each finiten,whosesupremumistheepsilonnumber, whose supremum is the epsilon number \varepsilon_0.Everyepsilonnumber. Every epsilon number \varepsilon_\alphaismultiplicativelyindecomposable,astheordinalsbelowitareclosedunderboth[addition](/page/Addition)and[multiplication](/page/Multiplication).[](https://core.ac.uk/download/pdf/82779240.pdf)Largerexamples,suchasis multiplicatively indecomposable, as the ordinals below it are closed under both [addition](/page/Addition) and [multiplication](/page/Multiplication).[](https://core.ac.uk/download/pdf/82779240.pdf) Larger examples, such as\omega^{\omega^{\varepsilon_0}}$, extend this pattern into the Veblen hierarchy. The class of multiplicatively indecomposable ordinals is itself closed under ordinal exponentiation, since raising one to a power preserves the indecomposability property through the exponential structure.

References

  1. https://mathworld.wolfram.com/LimitOrdinal.html
  2. https://stacks.math.columbia.edu/tag/05N1
  3. https://ncatlab.org/nlab/show/ordinal%2Bnumber
  4. https://bookofproofs.github.io/branches/set-theory/limit-ordinal.html
  5. http://math0.bnu.edu.cn/~shi/teaching/reading/Jech_ST_I_2in1.pdf
  6. https://builds.openlogicproject.org/content/set-theory/ordinals/opps.pdf
  7. https://people.maths.ox.ac.uk/knight/lectures/formalordinals.pdf
  8. https://neugierde.github.io/cantors-attic/Ordinal
  9. https://www.infinitelymore.xyz/p/well-ordinals-and-the-ordinal-numbers
  10. https://www.math.utoronto.ca/ivan/mat327/docs/notes/10-orders.pdf
  11. https://eprints.illc.uva.nl/id/eprint/232/1/PP-2006-57.text.pdf
  12. https://www.math.wustl.edu/~freiwald/ch8.pdf
  13. https://planetmath.org/ordinalspace
  14. https://jjsch.github.io/output/oa.pdf
  15. https://people.maths.ox.ac.uk/knight/lectures/moreordinals.pdf
  16. https://ncatlab.org/nlab/show/countable+ordinal
  17. https://boisgera.github.io/countable-ordinals/
  18. https://neugierde.github.io/cantors-attic/Epsilon_naught
  19. https://proofwiki.org/wiki/Limit_Ordinals_Preserved_Under_Ordinal_Multiplication
  20. https://proofwiki.org/wiki/Limit_Ordinals_Closed_under_Ordinal_Exponentiation
  21. https://www2.math.upenn.edu/~pemantle/DRP/04-transfinite.pdf
  22. https://builds.openlogicproject.org/content/set-theory/set-theory.pdf
  23. https://math.wvu.edu/~kciesiel/Other/ElectronicReprints/B2IntSetThe.pdf
  24. https://www.math.mcgill.ca/gsams/drp/papers/papers2019/2019Fall_Tao.pdf
  25. https://www.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf
  26. https://www2.math.uconn.edu/~solomon/math5026f18/OrdCard2.pdf
  27. https://stacks.math.columbia.edu/tag/000E
  28. https://www.math.mcgill.ca/makkai/SetTheory/SetTheoryPart2.pdf
  29. https://ncatlab.org/nlab/show/regular+cardinal
  30. https://ncatlab.org/nlab/show/inaccessible+cardinal
  31. https://planetmath.org/limitcardinal
  32. https://planetmath.org/additivelyindecomposable
  33. https://jdh.hamkins.org/tag/additively-indecomposable-ordinals/
  34. https://www.infinitelymore.xyz/p/how-to-count-to-infinity-and-beyond
  35. https://planetmath.org/veblenfunction
  36. https://core.ac.uk/download/pdf/82779240.pdf
  37. https://arxiv.org/pdf/1511.05882
  38. https://people.maths.bris.ac.uk/~mapdw/current-set-theory.pdf
  39. https://www.winterschool.eu/files/1466-On_the_binary_linear_ordering910828170.pdf
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