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Ordinal arithmetic

Ordinal arithmetic is a branch of set theory that extends the basic arithmetic operations—addition, multiplication, and exponentiation—to ordinal numbers, which represent the order types of well-ordered sets and generalize the natural numbers to transfinite quantities.^[1] These operations are defined recursively based on the structure of ordinals as transitive sets well-ordered by the membership relation, ensuring that the results remain ordinals and preserve well-ordering.^[2] Unlike arithmetic on finite numbers or cardinals (which focus on size and are commutative), ordinal arithmetic is sensitive to the order of operands and often yields asymmetric results, such as ω + 1 ≠ 1 + ω, where ω denotes the order type of the natural numbers.^[3] Ordinal numbers, or simply ordinals, form a hierarchy that captures the possible lengths of well-orderings, starting from the empty set (0), successors like 1 = {0}, and limit ordinals like ω, the least infinite ordinal.^[1] They are uniquely determined up to isomorphism by their order type, allowing arithmetic to be defined via concatenation of orderings: for addition, α + β is the order type of α followed by β; for multiplication, α · β replaces each element of β with a copy of α; and for exponentiation, α^β is the supremum of α^γ for γ < β at limits, or α^β · α for successors.^[2] These definitions ensure continuity at limit ordinals, where operations take the supremum over previous values.^[3] A defining feature of ordinal arithmetic is its non-commutativity and non-distributivity in certain directions: addition and multiplication are associative but addition is not commutative (e.g., n + ω = ω for finite n, but ω + n > ω), and multiplication distributes over addition from the left but not the right.^[1] Exponentiation follows similar recursive rules and satisfies α^(β + γ) = α^β · α^γ, but exhibits absorption properties like α · ω = sup{α · n | n < ω} = α ⋅ ω for infinite α.^[3] These properties distinguish ordinal arithmetic from cardinal arithmetic, where infinite operations often collapse to the maximum cardinality, and underpin advanced results in set theory, such as the construction of ordinal notations and proofs of consistency in impredicative systems.^[2]

Fundamentals

Ordinal numbers overview

Ordinal numbers, also known as ordinals, extend the concept of natural numbers to describe the order types of well-ordered sets in set theory. They are defined as equivalence classes of well-ordered sets, where two sets belong to the same class if there exists an order-isomorphism between them.^[4] This construction captures the structure of ordering rather than mere cardinality.^[5] The finite ordinals coincide with the natural numbers, represented as 0, 1, 2, and so forth.^[4] The smallest infinite ordinal, denoted \omega, corresponds to the order type of the natural numbers under their standard ordering.^[6] Ordinals are built via transfinite succession: a successor ordinal \alpha + 1 is obtained by adjoining a single element after all elements of \alpha, while a limit ordinal is the least upper bound (supremum) of an increasing sequence of smaller ordinals.^[7] Basic examples of ordinals include the finite ones 0, 1, 2, ..., followed by \omega, its successor \omega + 1, then \omega \cdot 2 (the order type of two copies of the naturals in sequence), and \omega^2 (the order type of \omega many copies of \omega).^[4] In contrast to cardinal numbers, which quantify the size or cardinality of sets without regard to order, ordinals emphasize the positional structure and well-ordering, enabling distinctions among sets of equal cardinality but different order types.^[8]

Set-theoretic construction

In set theory, the von Neumann construction defines each ordinal number \alpha as the set consisting of all ordinal numbers strictly less than \alpha, that is, \alpha = \{\beta \mid \beta < \alpha\}. This recursive definition begins with the empty set \emptyset as the ordinal 0, the successor ordinal 1 as \{0\} = \{\emptyset\}, 2 as \{0, 1\} = \{\emptyset, \{\emptyset\}\}, and continues transfinitely, ensuring that every ordinal is a transitive set whose elements are precisely the preceding ordinals. This approach embeds the ordinals directly within the cumulative hierarchy of sets, providing a concrete representation without relying on external equivalence classes of well-orderings. The order relation on these von Neumann ordinals is given by the membership relation: \alpha < \beta if and only if \alpha \in \beta. This defines a strict total order because the transitivity of ordinals ensures that if \alpha \in \beta and \beta \in \gamma, then \alpha \in \gamma, and the well-foundedness of the membership relation on ordinals prevents infinite descending chains. Equality of ordinals \alpha = \beta holds precisely when \alpha \subseteq \beta and \beta \subseteq \alpha, which, given the extensionality axiom of set theory, is equivalent to the standard set equality \forall \gamma (\gamma \in \alpha \leftrightarrow \gamma \in \beta). Since ordinals are transitive sets (every element is a subset), the subset relation aligns perfectly with the order, making the construction internally consistent.^[9]^[10] To verify that this construction produces sets isomorphic to the intended ordinals—equivalence classes of well-ordered sets up to order-isomorphism—one shows that the class of von Neumann ordinals, ordered by membership, is well-ordered and that every well-ordering on a set is order-isomorphic to a unique initial segment of this class. Specifically, for any well-ordered set (W, <), the order-type ordinal \alpha is the unique von Neumann ordinal such that W is order-isomorphic to \alpha, achieved by mapping each element to the ordinal of its predecessors. This isomorphism theorem follows from the Mostowski collapse lemma applied to the transitive closure, confirming that the von Neumann ordinals faithfully represent all possible well-order types without redundancy or gaps.^[9] The transfinite recursion theorem provides the foundational mechanism for defining functions and operations on ordinals by recursion along their well-ordering. It states that given a class X and a class function F: V \to V (where V is the universe of sets), there exists a unique function G: \mathrm{On} \to X (with \mathrm{On} the class of ordinals) such that for every ordinal \alpha,

If \alpha = 0, then G(0) is the base value (often specified separately),
If \alpha = \beta + 1 is a successor, then G(\alpha) = F(G(\beta)),
If \alpha is a limit ordinal, then G(\alpha) = \sup_{\beta < \alpha} G(\beta) or another limit rule defined by F.

This theorem is provable in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), relying on the replacement axiom to ensure the recursive definition yields a proper class function, and it underpins all subsequent definitions in ordinal theory, such as arithmetic operations.^[9]^[10]

Binary operations

Addition

Ordinal addition, denoted \alpha + \beta, is defined as the order type of the disjoint union of two well-ordered sets with order types \alpha and \beta, where the elements of the set corresponding to \beta are placed after those of \alpha. Specifically, if A and B are disjoint well-ordered sets with order types \alpha and \beta, respectively, then \alpha + \beta is the order type of A \cup B under the ordering where all elements of A precede all elements of B, preserving the internal orders of A and B.^[11] This operation admits a recursive definition that aligns with the structure of ordinals as successors or limits. For the zero ordinal, \alpha + 0 = \alpha. For a successor ordinal \beta + 1, \alpha + (\beta + 1) = (\alpha + \beta) + 1, which appends a single element after the structure of \alpha + \beta. For a limit ordinal \lambda, \alpha + \lambda = \sup\{\alpha + \mu \mid \mu < \lambda\}, taking the least upper bound of the additions with proper initial segments of \lambda. This recursive characterization ensures that ordinal addition respects the well-ordering principle and transfinite nature of ordinals.^[11]^[12] Illustrative examples highlight the behavior of addition with finite and infinite ordinals. Adding a finite ordinal to an infinite one yields an infinite result; for instance, $1 + \omega = \omega, as prepending a single element to the natural numbers does not alter the order type, since the successor of any finite initial segment remains countable in the same way. In contrast, \omega + 1 is strictly greater than \omega, as it appends an element after the entire infinite sequence, creating a new limit point. These cases demonstrate that finite ordinals added to infinite ones preserve the infinitude, but the placement matters significantly.^[11]^[12] A defining feature of ordinal addition is its non-commutativity: in general, \alpha + \beta \neq \beta + \alpha. The example \omega + 1 > 1 + \omega = \omega shows this asymmetry, as the order type depends on the sequence of concatenation—appending infinity after a finite element differs from the reverse. This contrasts with cardinal arithmetic, where addition is commutative, underscoring the ordinal focus on order types rather than mere size.^[11]^[12] Ordinal addition is associative: (\alpha + \beta) + \gamma = \alpha + (\beta + \gamma) for all ordinals \alpha, \beta, \gamma. This property holds by transfinite induction on \gamma, verifying the base case for zero, the successor case by appending one element, and the limit case via the supremum construction, ensuring the concatenated order types align regardless of grouping. Associativity facilitates the extension of addition to finite sums and underpins further developments in ordinal arithmetic.^[11]^[12]

Multiplication

Ordinal multiplication extends the concept of ordinal addition by defining the product \alpha \cdot \beta as the order type of the well-ordered set obtained by concatenating \beta many copies of a well-ordering of type \alpha. Set-theoretically, this is realized by taking the Cartesian product \beta \times \alpha equipped with the lexicographic order, where (b_1, a_1) < (b_2, a_2) if b_1 < b_2 or if b_1 = b_2 and a_1 < a_2, yielding a well-ordering whose order type is \alpha \cdot \beta.^[13]^[14] The operation is defined recursively on the second argument \beta, building upon ordinal addition:

\alpha \cdot 0 = 0,
\alpha \cdot (\beta + 1) = \alpha \cdot \beta + \alpha,
for a limit ordinal \lambda, \alpha \cdot \lambda = \sup \{ \alpha \cdot \mu \mid \mu < \lambda \}.^[12]^[14]

This recursive structure illustrates multiplication as repeated addition, where finite successors append additional copies of \alpha, and limits take the supremum of partial products. For instance, $2 \cdot \omega concatenates countably many copies of a 2-element order, resulting in an order type isomorphic to \omega, as the even and odd positions interleave into a single countable sequence. In contrast, \omega \cdot 2 concatenates two copies of \omega, yielding \omega + \omega, which is strictly larger than \omega. Further, \omega \cdot \omega arises as the limit of \omega \cdot n for finite n, forming \omega^2, a countable ordinal representing countably many countable sequences.^[12]^[13] Unlike cardinal multiplication, ordinal multiplication is non-commutative: in general, \alpha \cdot \beta \neq \beta \cdot \alpha. The example $2 \cdot \omega = \omega < \omega \cdot 2 = \omega + \omega demonstrates this, as the order type depends on the direction of concatenation—repeating a finite order countably many times absorbs into a limit, while repeating a limit finitely many times preserves the separation.^[12]^[14] Ordinal multiplication is associative: (\alpha \cdot \beta) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma) for all ordinals \alpha, \beta, \gamma, which follows by induction on \gamma using the recursive definition and the associativity of addition. It distributes over addition from the right: \alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma, provable similarly by cases on \gamma, but fails from the left: (\alpha + \beta) \cdot \gamma \neq \alpha \cdot \gamma + \beta \cdot \gamma in general. For example, (1 + 1) \cdot \omega = 2 \cdot \omega = \omega, whereas $1 \cdot \omega + 1 \cdot \omega = \omega + \omega > \omega.^[12]^[13]

Power operations

Exponentiation

Ordinal exponentiation is a binary operation on ordinal numbers α and β, denoted α^β, which extends the notion of finite exponentiation to the transfinite realm through repeated multiplication. It produces order types that grow superexponentially, far outpacing addition and multiplication, and is fundamental to representing large ordinals in set theory. The operation is defined recursively on the exponent β using transfinite recursion, ensuring continuity at limit stages.^[15] The recursive definition proceeds as follows:

\alpha^0 = 1

for any ordinal α (with the convention 0^0 = 1),

\alpha^{\beta+1} = \alpha^\beta \cdot \alpha

for successor ordinals β + 1, and

\alpha^\lambda = \sup \{ \alpha^\mu \mid \mu < \lambda \}

for limit ordinals λ > 0 (with α^λ = 0 if α = 0 and λ > 0). This construction leverages the previously defined ordinal multiplication and supremum operation.^[15]^[12] Equivalently, α^β can be understood set-theoretically as the order type of the set of all functions f: β → α with finite support (the domain where f(ξ) ≠ 0 is finite), ordered reverse-lexicographically: f < g if at the largest ξ where f(ξ) ≠ g(ξ), we have f(ξ) < g(ξ). This interpretation aligns with the Cantor normal form expansion of ordinals as "polynomials" in ω with coefficients less than the base.^[16] Illustrative examples highlight the operation's distinct behavior from finite arithmetic. For finite bases, 2^ω = \sup { 2^n \mid n < \omega } = \omega, as the union of increasing finite segments yields the order type of the naturals.^[17] Further, ω^2 = ω · ω, consisting of ω many copies of ω arranged successively. The tower ω^ω = \sup { ω^n \mid n < \omega } then captures an infinite iteration, yielding the least fixed point of α ↦ ω^α.^[17]^[18] Unlike cardinal exponentiation, ordinal exponentiation fails many familiar algebraic properties. It is non-commutative: for instance, 2^ω = ω < ω · ω = ω^2. More specifically, post-multiplying the result by γ does not generally equal exponentiating a product of exponents: α^β · γ ≠ α^(β · γ); taking α = ω, β = 2, γ = ω gives ω^2 · ω = ω^3 < ω^ω = ω^(2 · ω). This arises because multiplication absorbs into the exponent only under certain conditions, unlike the right-distributivity over addition α^(β + γ) = α^β · α^γ that does hold.^[12]^[17] Exponentiation also fails left-distributivity over multiplication: (α · β)^γ ≠ α^γ · β^γ in general. A concrete counterexample is (ω · 2)^2 = (ω · 2) · (ω · 2) = ω^2 · 2, while ω^2 · 2^2 = ω^2 · 4 > ω^2 · 2, since multiplying ω^2 on the right by finite ordinals preserves the leading term but accumulates more copies as the finite factor increases. These failures underscore the asymmetry inherent in ordinal order types, where left-to-right concatenation dominates.^[12]^[18]

Tetration and higher

Tetration extends ordinal exponentiation to iterated exponentiation, forming towers of exponents of height given by an ordinal. The operation is denoted {}^\beta \alpha, where \alpha is the base and \beta is the height, defined recursively on ordinals. For the base case, {}^0 \alpha = 1. For successor ordinals, {}^{\beta+1} \alpha = \alpha^{({}^\beta \alpha)}, using ordinal exponentiation. At limit ordinals \lambda, {}^\lambda \alpha = \sup \{ {}^\gamma \alpha \mid \gamma < \lambda \}. This recursive extension aligns with the broader hyperoperation sequence, where tetration corresponds to the fourth level after addition, multiplication, and exponentiation. On finite ordinals, it recovers the standard tetration from natural number arithmetic. For example, the infinite tetration {}^\omega \omega is the least ordinal \varepsilon_0 satisfying \omega^{\varepsilon_0} = \varepsilon_0, the first epsilon number. Further growth occurs with higher heights; for instance, {}^{\omega+1} \omega = \omega^{({}^\omega \omega)} = \omega^{\varepsilon_0} = \varepsilon_0, but iterating to {}^\omega \varepsilon_0 yields the next epsilon number \varepsilon_1.^[19] Pentation, the fifth hyperoperation, iterates tetration similarly: {}^\beta \alpha at level 5 is a tower of tetrations of height \beta. Higher operations follow analogously in the sequence, distributing over Cantor normal form for computability. These Ackermann-like functions on ordinals extend the rapid growth of the Ackermann function on naturals but encounter ordinal-specific convergence via suprema at limits, ensuring well-definedness within the class of ordinals while stabilizing at fixed points like epsilon numbers.^[20] In contrast to finite hyperoperations, which grow without bound on naturals, ordinal versions incorporate absorption properties, such as {}^\beta \alpha = \alpha for sufficiently large \beta when \alpha is an epsilon number, highlighting the role of ordinal limits in bounding vertical iteration.^[20]

Structural representations

Cantor normal form

The Cantor normal form provides a unique way to express every nonzero ordinal as a finite sum resembling a polynomial in the base \omega, where \omega is the smallest infinite ordinal. Specifically, for any ordinal \alpha > 0, there exist ordinals \beta_k > \beta_{k-1} > \cdots > \beta_0 and positive finite ordinals (natural numbers) $1 \leq c_i < \omega such that

\alpha = \omega^{\beta_k} \cdot c_k + \omega^{\beta_{k-1}} \cdot c_{k-1} + \cdots + \omega^{\beta_0} \cdot c_0,

with the sum interpreted in the ordinal sense (left-to-right addition). This representation leverages the fact that every ordinal is either a successor or a limit, and powers of \omega capture the "principal" terms in the decomposition.^[21]^[22] The uniqueness of this form, known as Cantor's normal form theorem, follows from transfinite induction on \alpha and the strict decreasing order of the exponents. To construct the form via a greedy algorithm, identify the largest ordinal \beta_k such that \omega^{\beta_k} \leq \alpha, then determine the largest finite c_k with \omega^{\beta_k} \cdot c_k \leq \alpha, and recurse on the remainder \alpha - \omega^{\beta_k} \cdot c_k < \omega^{\beta_k}. Uniqueness arises because any two such representations must agree on the leading term (highest exponent), as differing leading exponents would violate the size comparison via ordinal addition properties; subsequent terms are then uniquely determined by induction. This process terminates due to the well-foundedness of the ordinals.^[21]^[22]^[23] Representative examples illustrate the form's structure. For instance, \omega + 3 = \omega^1 \cdot 1 + \omega^0 \cdot 3, where \omega^1 = \omega and \omega^0 = 1. A more advanced case is the least fixed point of the exponentiation function \alpha \mapsto \omega^\alpha, denoted \varepsilon_0, which satisfies \varepsilon_0 = \omega^{\varepsilon_0}. These expressions highlight how finite coefficients adjust the "multiplicity" of each power, while the decreasing exponents ensure no redundancy.^[22]^[23] Arithmetic operations can be performed directly on ordinals in Cantor normal form, treating the expressions like polynomials but respecting ordinal non-commutativity. For addition \alpha + \beta, align the terms by their exponents (padding with zeros if needed), add coefficients for matching exponents, and propagate any "carry-over" if a coefficient reaches \omega (replacing \omega \cdot \omega^\gamma with \omega^{\gamma+1}). For example, (\omega^2 \cdot 3 + \omega) + (\omega^2 \cdot 2 + 1) = \omega^2 \cdot 5 + \omega + 1. Multiplication \alpha \cdot \beta follows distributive rules: expand \beta's terms and multiply each by \alpha, then combine like terms with carry-over, using \omega^\gamma \cdot m = \omega^\gamma \cdot (m-1) + \omega^\gamma for finite m and \omega^{\beta} \cdot \omega^{\delta} = \omega^{\beta + \delta}. An example is (\omega^3 \cdot 2 + 2) \cdot (\omega + 5) = \omega^4 + \omega^3 \cdot 15 + \omega \cdot 10 + 10. These algorithms preserve the normal form after normalization.^[21]^[23] Ordinal comparison is facilitated by the normal form through a lexicographic order on the sequences of coefficients, starting from the highest exponent. For two ordinals \alpha = \omega^{\beta_m} \cdot c_m + \cdots and \alpha' = \omega^{\beta'_n} \cdot c'_n + \cdots with \beta_m = \beta'_n, the first position i where c_i \neq c'_i determines the order: \alpha < \alpha' if c_i < c'_i at that position, due to the absorption properties of higher terms over lower ones. If the leading exponents differ, the one with the larger leading exponent is greater. This method underpins the uniqueness proof and efficient decision procedures for ordinal inequalities.^[21]^[22]

Prime factorization

In ordinal arithmetic, a prime ordinal is defined as an ordinal greater than 1 that is indecomposable under multiplication, meaning it cannot be expressed as the product of two ordinals both strictly between 1 and itself.^[24] Such primes include the finite primes (2, 3, 5, ...) as well as infinite examples that are additively closed, such as \omega, \omega^\omega, and \varepsilon_0, the least fixed point of the exponential function \alpha \mapsto \omega^\alpha.^[23] The fundamental theorem of ordinal prime factorization, established by Ernst Jacobsthal, states that every ordinal \alpha > 1 admits a unique factorization as a finite product \alpha = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}, where the p_i are distinct prime ordinals and the exponents e_i \geq 1 are positive ordinals, with uniqueness holding for the minimal factorization, where limit primes precede successor primes, primes within each type are in non-increasing order, and finite prime factors are unique up to reordering.^[24] This canonical form ensures that non-commutativity and absorption properties (e.g., n \cdot \omega = \omega for finite n > 1) do not undermine uniqueness, by imposing strict ordering on the factors. For instance, the ordinal \omega^2 + \omega factors as \omega \cdot (\omega + 1), where both \omega (limit prime) and \omega + 1 (successor prime) are primes; the canonical form respects the ordering with the limit prime preceding the successor prime.^[23] Similarly, \varepsilon_0 itself is prime, as it cannot be decomposed further under multiplication.^[23] To compute this factorization, one may leverage the Cantor normal form of the ordinal, which expresses \alpha = \omega^{\beta_1} \cdot c_1 + \omega^{\beta_2} \cdot c_2 + \cdots + \omega^{\beta_m} \cdot c_m with \beta_1 > \beta_2 > \cdots > \beta_m \geq 0 and finite coefficients c_i \geq 1. The prime powers are extracted by recursively factoring the exponents \beta_i and the coefficients c_i into their ordinal primes, then reassembling the product while preserving the decreasing order of primes; this process relies on the normal form theorem for its validity.^[25] Standard multiplication's non-commutativity can lead to multiple representations, resolved by the canonical ordering; for broader uniqueness in commutative settings, connections to Hessenberg natural sum and product operations are necessary, which symmetrize the arithmetic.^[24]

Advanced properties

Absorption and limits

In ordinal arithmetic, absorption laws describe situations where adding, multiplying, or exponentiating a smaller ordinal to a larger infinite one yields the larger ordinal unchanged. For addition, if α is an infinite limit ordinal and β < α, then β + α = α. This right absorption property holds because the order type of β followed by α merges β into the initial segment of α without altering its overall structure, as α has no maximum element. Similar absorption occurs in multiplication and exponentiation for certain infinite ordinals; for instance, if α is an infinite limit ordinal and β ≥ 1 is finite, then α · β = α · (β - 1) + α, but the full absorption like α · ω = sup_{n < ω} (α · n) equals α only under specific conditions, such as when α is additively closed. A representative example is n · η = η for any finite ordinal n ≥ 1 and limit ordinal η ≥ ω, where the finite repetitions are absorbed into the limit structure of η. In exponentiation, ω^ω absorbs lower terms, as ω^ω = sup_{n < ω} ω^n, effectively subsuming finite powers into the limit. At limit ordinals, ordinal operations exhibit continuity from below, defined via suprema of preceding values. For a limit ordinal γ, the sum α + γ is the supremum of {α + β | β < γ}, ensuring the operation respects the well-ordered nature without a final successor step. Similarly, for multiplication, α · γ = sup_{β < γ} (α · β), and for exponentiation, α^γ = sup_{β < γ} (α^β). This supremum-based definition preserves the order type at limits, making ordinal arithmetic continuous in the right argument for these operations. For example, ω · ω = sup_{n < ω} (ω · n) = sup_{n < ω} ω · n, which equals ω · ω, illustrating how the limit captures the growing sequence without collapse. Ordinal operations are monotonic, meaning they preserve inequalities in a controlled manner. Specifically, if α ≤ α' and β ≤ β', then α + β ≤ α' + β', α · β ≤ α' · β', and α^β ≤ (α')^{β'} under suitable conditions like β > 0. Strict monotonicity holds in the right argument: if α < β, then γ + α < γ + β for any γ, and similarly for multiplication and exponentiation when the left operand is positive. However, left strictness requires additional assumptions, such as the right operand being nonzero. These properties follow from the recursive definitions and the well-ordering of ordinals, ensuring operations do not decrease order types. Unlike cardinal arithmetic, where infinite cardinals are idempotent (κ + κ = κ), ordinal addition fails idempotence for infinite ordinals: α + α > α. This occurs because the second copy of α is appended after the first, creating a distinct order type larger than α, as seen in ω + ω = ω · 2 > ω. The failure stems from the non-commutative, order-sensitive nature of ordinal operations, where position matters, preventing the absorption of the second α into the first.

Large countable ordinals

The Veblen hierarchy provides a foundational method for constructing large countable ordinals through a transfinite sequence of normal functions \phi_\alpha: \mathrm{Ord} \to \mathrm{Ord}, where each \phi_\alpha enumerates fixed points of preceding functions in the hierarchy. Introduced by Oswald Veblen in his study of continuous increasing functions on ordinals, the hierarchy begins with \phi_0(\beta) = \omega^\beta for any ordinal \beta, corresponding to ordinal exponentiation with base \omega. Successor levels are defined by letting \phi_{\gamma+1} enumerate the fixed points of \phi_\gamma, i.e., \phi_{\gamma+1}(\beta) is the \beta-th ordinal \xi such that \phi_\gamma(\xi) = \xi, while at limit ordinals \lambda, \phi_\lambda(\beta) is the pointwise supremum of \phi_\alpha(\beta) for \alpha < \lambda. This recursive construction generates increasingly large ordinals while remaining within the countable realm, as all such \phi_\alpha(\beta) are strictly less than the first uncountable ordinal \omega_1. A prominent example is the ordinal \varepsilon_0 = \phi_1(0), the least fixed point of the exponentiation function \alpha \mapsto \omega^\alpha, equivalently the supremum of the sequence \omega, \omega^\omega, \omega^{\omega^\omega}, \dots obtained as the limit of finite tetrations of \omega. This ordinal, an \varepsilon-number, is closed under ordinal addition and multiplication, meaning that for all \beta, \gamma < \varepsilon_0, \beta + \gamma < \varepsilon_0 and \beta \cdot \gamma < \varepsilon_0, highlighting its role in absorbing smaller ordinals under these operations. Further along the hierarchy, \zeta_0 = \phi_2(0) denotes the least fixed point of the \varepsilon-mapping, i.e., the smallest \zeta such that \varepsilon_\zeta = \zeta, forming a \zeta-number with similar closure properties extended to the \varepsilon-numbers, meaning that for all \beta, \gamma < \zeta_0, \beta + \gamma < \zeta_0, \beta \cdot \gamma < \zeta_0, and \varepsilon_\beta < \zeta_0. The progression continues to \phi_\omega(0) = \Gamma_0, the Feferman–Schütte ordinal, which is the least upper bound of \{\phi_n(0) \mid n < \omega\} and the first ordinal closed under the entire finite-argument Veblen functions, marking a significant milestone in hierarchies of ordinal arithmetic. The large Veblen ordinal, often denoted \vartheta(\varepsilon_0) or realized as \phi_{\varepsilon_0}(0), represents the supremum of all terms \phi_\alpha(\beta) where \alpha, \beta < \varepsilon_0, serving as the first fixed point of the diagonal function \alpha \mapsto \phi_\alpha(0) and thus the culmination of the standard (one-variable) Veblen hierarchy. This ordinal encapsulates all countable ordinals definable via finite iterations of Veblen functions up to \varepsilon_0 arguments, remaining additively, multiplicatively, and exponentially closed with respect to the sub-hierarchy below it. Beyond this, the Bachmann–Howard ordinal arises as the limit of an extended hierarchy incorporating ordinal notations with uncountable indices relative to a Mahlo cardinal analogue in the countable setting, specifically \phi_{\varepsilon_\Omega + 1}(0) where \Omega denotes the least ordinal greater than all previous notations, providing a bound for more advanced constructive systems while still countable. These ordinals illustrate how arithmetic operations like exponentiation and tetration, iterated hierarchically, yield structures closed under prior levels of the hierarchy, enabling the systematic enumeration of vast segments of the countable ordinals.^[26]^[27]

Extensions and applications

Natural sum and product

The natural sum and natural product of ordinals are commutative operations that provide symmetric analogues to the standard non-commutative ordinal addition and multiplication, mimicking the behavior of cardinal arithmetic in a well-ordered context. These operations, also known as the Hessenberg sum and Hessenberg product, were introduced by Gerhard Hessenberg in 1906 and later axiomatized by Philip W. Carruth in 1942 as the unique smallest operations satisfying certain natural properties.^[28] They are particularly useful for computations involving Cantor normal forms and extend finite arithmetic consistently to transfinite ordinals. The natural sum \alpha \oplus \beta is defined using the Cantor normal forms of \alpha and \beta. Write \alpha = \sum_{i=1}^m \omega^{\xi_i} \cdot c_i and \beta = \sum_{j=1}^n \omega^{\eta_j} \cdot d_j, where \xi_1 > \cdots > \xi_m \geq 0, \eta_1 > \cdots > \eta_n \geq 0, and each c_i, d_j is a positive finite ordinal. Align the expressions over the common decreasing sequence of distinct exponents from \{\xi_1, \dots, \xi_m\} \cup \{\eta_1, \dots, \eta_n\}, padding missing coefficients with 0. Then, add the coefficients componentwise to obtain \sum_k \omega^{\zeta_k} \cdot e_k, where each e_k = c_k + d_k < \omega, and rewrite in standard Cantor normal form if necessary (though no carrying occurs for finite coefficients). This yields \alpha \oplus \beta. If the leading exponent of \alpha exceeds that of \beta, the leading term of \alpha \oplus \beta matches that of \alpha, so \alpha absorbs \beta in the sense that lower-order terms from \beta do not affect the dominant structure.^[29] The natural sum is commutative (\alpha \oplus \beta = \beta \oplus \alpha) and associative (\alpha \oplus (\beta \oplus \gamma) = (\alpha \oplus \beta) \oplus \gamma). It is also monotonic: \alpha \leq \beta implies \gamma \oplus \alpha \leq \gamma \oplus \beta. Unlike the standard ordinal sum, it is not continuous in the right argument and does not satisfy \alpha \oplus \alpha = \alpha in general (though it does for finite \alpha). For example, \omega \oplus 1 = \omega + 1, since \omega = \omega^1 \cdot 1 and $1 = \omega^0 \cdot 1, yielding coefficients $1for\omega^1

and &#36;1

for \omega^0. Similarly, \omega \oplus \omega = \omega \cdot 2.^[28]^[29] The natural product \alpha \otimes \beta is the commutative counterpart to ordinal multiplication, computed as the "polynomial product" in base \omega. Using the normal forms above, \alpha \otimes \beta = \sum_{i,j} c_i \cdot d_j \cdot \omega^{\xi_i + \eta_j}, where the right-hand side is expanded, like terms (same exponents) have coefficients added, and the result is normalized to Cantor normal form. This aligns with the sum of products of corresponding terms and ensures the operation respects the well-ordering. If one ordinal has a strictly higher leading exponent, its influence dominates the product's structure, analogous to absorption in the sum.^[29] The natural product is commutative (\alpha \otimes \beta = \beta \otimes \alpha), associative (\alpha \otimes (\beta \otimes \gamma) = (\alpha \otimes \beta) \otimes \gamma), and distributes over the natural sum: \alpha \otimes (\beta \oplus \gamma) = (\alpha \otimes \beta) \oplus (\alpha \otimes \gamma). It has 1 as the identity: \alpha \otimes 1 = \alpha. For example, \omega \otimes 2 = \omega \cdot 2, since $2 = \omega^0 \cdot 2 and the product expands to \omega^1 \cdot 2 = \omega \cdot 2. Another instance is \omega \otimes \omega = \omega^2. These properties make the structure (\mathrm{Ord}, \oplus, \otimes, 0, 1) a commutative semiring.^[28]^[29]

Nimber arithmetic

In combinatorial game theory, nimbers represent the equivalence classes of impartial games under disjunctive sum, with each nimber assigned a value via the Sprague-Grundy theorem, where the nimber of a position is the minimum excludant (mex) of the nimbers of its options. For the game of Nim, a nim-heap of finite size n, denoted *n, has nimber n under nim-addition, which coincides with the bitwise XOR operation for natural numbers. Ordinal nimbers generalize this to transfinite cases, where *\alpha for an ordinal \alpha is the impartial game with options *\beta for all \beta < \alpha, yielding nimber \alpha itself as its Grundy number. The disjunctive sum of two such games, *\alpha + *\beta, has nimber \alpha \oplus \beta, where \oplus denotes nim-addition on ordinals, defined by expressing \alpha and \beta in Cantor normal form with base 2 (as sums of distinct powers of 2) and taking the symmetric difference of their terms. For finite ordinals, this reduces to the standard XOR: *n + *m = *(n \oplus m). A key example is the infinite nim-heap *\omega, which has options *n for all finite n, resulting in nimber \omega = \ mex\{0, 1, 2, \dots \}. Nim-multiplication on ordinals, denoted \cdot or \circ, is defined recursively as the mex of all products of the form \alpha' \circ \beta, \alpha \circ \beta', and \alpha' \circ \beta' for proper initial segments \alpha' < \alpha and \beta' < \beta. This operation arises from the convolution product of impartial games, where the product game allows moves in one component followed by a response in the other. The structure (\mathsf{On}, \oplus, \circ) forms an algebraically closed field of characteristic 2, known as the nim-field \mathsf{On}_2, with \omega serving as its first transcendental element over the prime field.^[30] Nimber arithmetic connects to broader game values through surreal numbers, as ordinals embed naturally into the surreals as positive infinitesimal-free elements, allowing impartial games like nim-heaps to be analyzed within the surreal framework while preserving their algebraic properties under disjunctive sum and convolution.^[30]

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