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Epsilon number

In , an epsilon number is a transfinite ordinal \varepsilon that satisfies \varepsilon = \omega^\varepsilon, where \omega is the smallest infinite ordinal and is performed using . The concept was introduced by in his foundational work on transfinite numbers during the late , particularly in developing and normal forms for representing ordinals. The smallest epsilon number, denoted \varepsilon_0, is the least upper bound (supremum) of the sequence \omega, \omega^\omega, \omega^{\omega^\omega}, \dots, and it marks the boundary beyond which Cantor's normal form notation for ordinals requires extension. Epsilon numbers form a proper of ordinals that is closed under the operation \alpha \mapsto \varepsilon_\alpha, where \varepsilon_\alpha is the \alpha-th epsilon number in the canonical enumeration, and this class is unbounded in the ordinals, meaning for every ordinal \beta, there exists an epsilon number greater than \beta. Notably, every uncountable infinite cardinal \kappa is itself an epsilon number, since \kappa = \omega^\kappa in , highlighting their significance in the hierarchy of large ordinals and their applications in and descriptive set theory.

Definition and history

Definition in ordinal arithmetic

Epsilon numbers are ordinals \varepsilon satisfying \varepsilon = \omega^\varepsilon, where \omega is the least infinite ordinal and ^ denotes ordinal exponentiation. These represent fixed points of the function \alpha \mapsto \omega^\alpha. Ordinal exponentiation with base \omega is defined recursively: \omega^0 = 1, \omega^{\alpha+1} = \omega^\alpha \cdot \omega for successor ordinals, and for limit ordinals \lambda, \omega^\lambda = \sup \{ \omega^\alpha \mid \alpha < \lambda \}. The least epsilon number \varepsilon_0 is constructed as the supremum of the sequence obtained by iterating the exponential: let \alpha_0 = \omega and \alpha_{n+1} = \omega^{\alpha_n} for finite n < \omega; then \varepsilon_0 = \sup \{ \alpha_n \mid n < \omega \}. This yields \varepsilon_0 = \sup \{ \omega, \omega^\omega, \omega^{(\omega^\omega)}, \dots \}, satisfying \varepsilon_0 = \omega^{\varepsilon_0}. Equivalently, \varepsilon_0 = \sup \{ \omega^{\varepsilon_0 \cdot n} \mid n < \omega \}. The \alpha-th epsilon number \varepsilon_\alpha is enumerated by transfinite recursion as follows: \varepsilon_0 as defined above. For a successor ordinal \beta + 1, define a sequence \delta_0 = \varepsilon_\beta + 1 and \delta_{n+1} = \omega^{\delta_n} for n < \omega; then \varepsilon_{\beta+1} = \sup \{ \delta_n \mid n < \omega \}. For a limit ordinal \lambda, \varepsilon_\lambda = \sup \{ \varepsilon_\beta \mid \beta < \lambda \}. This enumerates the epsilon numbers in strictly increasing order.

Historical development

The concept of epsilon numbers originated in the late 19th century as part of Georg Cantor's foundational work on transfinite ordinals. In his 1897 paper, Cantor introduced \varepsilon_0 as the least ordinal fixed point of the exponential function, defined as the supremum of the tower \omega, \omega^\omega, \omega^{\omega^\omega}, and so on, thereby extending the hierarchy of countable ordinals beyond simple powers of \omega. This construction addressed limitations in earlier ordinal notations and highlighted the role of fixed points in transfinite arithmetic. In 1906, Gerhard Hessenberg advanced the formalization of ordinal notations in his treatise Grundbegriffe der Mengenlehre, where he defined operations such as the natural sum and product on ordinals, facilitating systematic representations up to epsilon numbers. Hessenberg's work built on Cantor's framework by introducing concepts like gamma numbers (additively closed ordinals) and delta numbers (multiplicatively closed), which are precursors to the epsilon numbers as fixed points under exponentiation. Oswald Veblen contributed significantly around 1908 with his development of ordinal hierarchies through continuous increasing functions, as detailed in his paper on transfinite ordinals. Veblen's phi functions provided a systematic enumeration of epsilon numbers, placing them within broader structures of normal functions and influencing subsequent ordinal arithmetic. The epsilon numbers gained prominence in proof theory through Gerhard Gentzen's 1936 consistency proof for Peano arithmetic, where he employed transfinite induction up to \varepsilon_0 to establish the system's consistency relative to this ordinal, advancing David Hilbert's formalist program. Following Gentzen, major developments remained limited until the 1960s, when proof-theoretic ordinal analysis by figures like Gaisi Takeuti and Solomon Feferman applied epsilon numbers to assess the strength of impredicative systems, though extensions beyond \varepsilon_0 received sparse attention in foundational work.

Ordinal epsilon numbers

Construction and enumeration

The epsilon numbers \varepsilon_\alpha for ordinals \alpha form the sequence of all solutions to the equation \xi = \omega^\xi, ordered increasingly by the standard ordinal ordering, with \varepsilon_0 being the least such solution. This enumeration ensures that each \varepsilon_\alpha is the \alpha-th epsilon number in this class, and the sequence is strictly increasing. The construction of the epsilon function \varepsilon(\alpha) proceeds via transfinite recursion on the normal function f(\xi) = \omega^\xi, which is continuous and strictly increasing on the ordinals. Specifically, \varepsilon(\alpha) is defined as the \alpha-th fixed point of f, starting from \varepsilon(0) as the least fixed point. For successor ordinals \alpha = \beta + 1, \varepsilon(\alpha) is the least ordinal strictly greater than \varepsilon(\beta) that satisfies the fixed-point equation. For limit ordinals \lambda, \varepsilon(\lambda) = \sup\{\varepsilon(\beta) \mid \beta < \lambda\}, ensuring continuity of the enumeration. For finite indices beyond 0, explicit constructions illustrate the process. The ordinal \varepsilon_1 is the least fixed point greater than \varepsilon_0, given by \varepsilon_1 = \sup\{\varepsilon_0, \varepsilon_0^{\varepsilon_0}, \varepsilon_0^{\varepsilon_0^{\varepsilon_0}}, \dots \}, where the supremum is taken over finite power towers of \varepsilon_0 (right-associated exponentiation). Similarly, \varepsilon_2 is the least fixed point greater than \varepsilon_1, obtained analogously as the supremum of finite power towers starting from \varepsilon_1; this pattern continues for each finite successor index. This recursive enumeration yields epsilon numbers \varepsilon_\alpha for every ordinal \alpha, forming a proper class of ordinals rather than a set, as the class of all fixed points of f is unbounded in the ordinals.

Properties

All epsilon numbers \varepsilon_\alpha are limit ordinals. The cofinality of \varepsilon_\alpha equals the cofinality of \alpha. In ordinal arithmetic, \varepsilon_\alpha + \beta = \varepsilon_\alpha for all \beta < \varepsilon_\alpha, reflecting their role as fixed points that absorb smaller ordinals under addition. However, \varepsilon_\alpha \cdot 2 = \varepsilon_\alpha + \varepsilon_\alpha > \varepsilon_\alpha, demonstrating that epsilon numbers are not idempotent under doubling. The of \varepsilon_\alpha aligns with that of its \alpha: \varepsilon_\alpha is countable whenever \alpha is countable, while uncountable epsilon numbers arise for uncountable indices. In , \varepsilon_0 is the proof-theoretic ordinal of Peano arithmetic (PA), serving as the supremum of ordinals for which suffices to establish the consistency of PA, as demonstrated by Gentzen's seminal consistency proof. Epsilon numbers up to certain points contribute to larger ordinal hierarchies, such as the Bachmann-Howard ordinal, which marks a significant extension in notations beyond the initial epsilon sequence. No epsilon number \varepsilon_\alpha equals \omega^\beta for any \beta < \varepsilon_\alpha, since this would imply \omega^\beta = \varepsilon_\alpha = \omega^{\varepsilon_\alpha} > \omega^\beta, yielding a contradiction.

Role in ordinal hierarchies

Veblen hierarchy

The Veblen hierarchy is a transfinite hierarchy of normal functions on the class of ordinal numbers, introduced by in his paper on continuous increasing functions of finite and transfinite ordinals, where he developed the theory of such functions and their fixed points to extend beyond basic arithmetic operations like and . These functions provide a systematic way to enumerate increasingly large ordinals, with epsilon numbers arising as the outputs of the second level in this hierarchy. The base function in the Veblen hierarchy is defined as \phi_0(\alpha) = \omega^\alpha for any ordinal \alpha, which enumerates the additively closed ordinals. The next level, \phi_1(\alpha), enumerates the fixed points of \phi_0, meaning \phi_1(\alpha) is the \alpha-th ordinal \varepsilon_\alpha such that \omega^{\varepsilon_\alpha} = \varepsilon_\alpha. Thus, the epsilon numbers \varepsilon_\alpha are precisely given by \varepsilon_\alpha = \phi_1(\alpha). Higher levels of the are defined recursively: for a successor ordinal \beta = \gamma + [1](/page/1), \phi_\beta(\alpha) enumerates the common fixed points of all \phi_\delta for \delta < \beta; for a limit ordinal \lambda, the function \phi_\beta at \lambda is the least upper bound, specifically \phi_\beta(\lambda) = \sup\{\phi_\beta(\gamma) \mid \gamma < \lambda\}. This construction ensures each \phi_\beta is normal (strictly increasing and continuous), and higher \phi_\beta for \beta > [1](/page/1) generate fixed points beyond the epsilons, such as the zeta numbers at \phi_2 and further iterations. Every ordinal below the \Gamma_0 = \phi_\omega(0) admits a unique representation in terms of the Veblen functions up to \phi_\beta for \beta < \omega, known as the Veblen normal form, which generalizes the Cantor normal form by incorporating these higher operations. Epsilon numbers, as fixed points of exponentiation, serve as the foundational layer in this hierarchy, enabling the enumeration of ordinals that surpass simple exponential towers.

Relation to larger fixed points

The epsilon numbers form an initial segment in the broader Veblen hierarchy, where the supremum of the sequence \varepsilon_\alpha for all \alpha < \Gamma_0 yields \Gamma_0 = \varphi_\omega(0), the first ordinal that is a fixed point of the epsilon function in the sense of being invariant under the full finite Veblen progression up to \omega. This \Gamma_0, known as the , marks the boundary beyond which the two-argument Veblen functions no longer suffice to enumerate all smaller ordinals without diagonalization. Zeta numbers \zeta_\alpha extend this structure as the fixed points of the epsilon function itself, satisfying \zeta_\alpha = \varepsilon_{\zeta_\alpha}, thereby enumerating the ordinals closed under the operation \alpha \mapsto \varepsilon_\alpha. These zeta numbers initiate higher levels in extended ordinal hierarchies, such as the , where successive fixed-point enumerations build toward stronger closure properties. In proof theory, epsilon numbers delineate the consistency strength of subsystems of second-order arithmetic; for instance, the subsystem \mathrm{ACA}_0 has proof-theoretic ordinal \varepsilon_0, bounding the ordinals representable in its cut-elimination process, while stronger subsystems like \mathrm{ATR}_0 reach up to \Gamma_0. This role highlights how epsilon numbers constrain the transfinite inductions admissible in formal systems capable of representing arithmetic truths. The class of epsilon numbers is closed under the successor operation in its enumeration—meaning \varepsilon_{\alpha+1} is always an epsilon number—but not under arbitrary limits unless the index set's supremum aligns with the function's continuity, necessitating higher enumerators like for limit closures. For very large indices, such as \varepsilon_{\omega_1}, this yields the first uncountable epsilon number, which has cofinality \omega_1 due to the normalcy of the epsilon function preserving cofinalities at limits.

Representations of ε₀

Tree representations

Ordinals less than ε₀ can be represented using finite rooted trees under a specific well-ordering, such as the one defined in the , a well-ordering on these trees where the height of a tree corresponds to the nesting depth of exponentiation in the ordinal's . In this system, each finite rooted tree encodes an ordinal through recursive assignment, with the structure reflecting the additive and exponential composition of smaller ordinals. The order ensures that every descending sequence of trees terminates, mirroring the well-foundedness of the ordinals below ε₀. The correspondence between trees and ordinals is given by interpreting the tree's structure in terms of powers of ω. The ordinal assigned to a tree T is defined recursively: if T is a single node, ord(T) = 0; otherwise, ord(T) = \sum_{i=1}^k \omega^{\mathrm{ord}(S_i)}, where S_1, \dots, S_k are the subtrees attached to the root, ordered by decreasing ord(S_i). This captures the exponential growth up to ε₀ as the supremum of all such finite structures. For instance, the tree consisting of only the root represents 0; a root with a single direct child (leaf) represents 1; a root with a chain root—internal node—leaf represents ω; and a suitable tree with a subtree of ordinal ω represents ω^ω, illustrating how additional levels introduce higher exponentiation. This representation allows for a direct visualization of ordinal arithmetic, where combining subtrees corresponds to addition and exponentiation in the normal form. This tree system is intimately connected to the hydra game, where finite rooted trees serve as hydras, and battles involve cutting heads, leading to regrowth that simulates ordinal descent. The Kirby-Paris theorem states that every hydra battle terminates in a number of steps ordinal-less-than ε₀, establishing the game's finite length despite apparent immortality. This result links directly to Goodstein's theorem on the termination of Goodstein sequences, both being unprovable in Peano arithmetic (PA) but true under transfinite induction up to ε₀, highlighting the proof-theoretic strength required beyond PA. The tree representation proves the well-foundedness of ordinals less than ε₀ through reductions: each valid move in the hydra game or embedding in the ordering decreases the associated ordinal, ensuring no infinite descending chains exist. This structural proof via tree transformations provides a concrete combinatorial witness to the ordinal's order type.

Other models

One alternative representation for ordinals less than \varepsilon_0 is the extended Cantor normal form, which expresses such ordinals as finite sums of the form \omega^{\beta_1} \cdot n_1 + \cdots + \omega^{\beta_k} \cdot n_k, where each \beta_i < \varepsilon_0, the \beta_i are strictly decreasing, and the n_i are positive finite ordinals (natural numbers). This form provides a unique decomposition, allowing explicit manipulation and comparison of these ordinals through operations on their exponents and coefficients. In proof theory, the slow-growing hierarchy offers a functional model for ordinals up to \varepsilon_0, where functions g_{\alpha}(n) for \alpha < \varepsilon_0 are provably total in (PA), while g_{\varepsilon_0}(n) grows faster than any function provably total in PA. These hierarchies classify the computational strength of PA, with g_{\varepsilon_0}(n) exceeding the bound of functions whose totality PA can prove. Multiplicative notation systems, such as those developed in proof-theoretic ordinal analysis (e.g., Beklemishev's multiplicative system or extensions in Rathjen's frameworks), approximate \varepsilon_0 by encoding ordinals through multiplicative principles and collapsing functions, facilitating consistency proofs for arithmetic. These notations emphasize multiplicative closure and are particularly useful for analyzing subsystems of analysis beyond basic addition and exponentiation. Unlike notations for higher epsilon numbers, which often require impredicative definitions or large cardinal assumptions, these models—Cantor normal forms, slow-growing hierarchies, and multiplicative systems—enable explicit computation and enumeration of ordinals up to \varepsilon_0 within recursive frameworks. Goodstein sequences provide a computational link to \varepsilon_0, as each sequence starting from a natural number terminates after fewer than \varepsilon_0 steps when interpreted in ordinal base representations, yet the theorem stating universal termination is unprovable in PA. This independence, established via ordinal descent arguments, highlights \varepsilon_0 as the proof-theoretic ordinal of PA.

Surreal epsilon numbers

Definition

The surreal numbers form a proper class extending the ordered field of real numbers to incorporate infinitesimal and transfinite quantities, introduced by John Horton Conway as a recursive construction where each surreal is born on a specific ordinal day and admits a unique normal form representation as a (possibly transfinite) sum of terms of the form n \cdot \omega^y with integer coefficients n and surreal exponents y in decreasing order. Surreal epsilon numbers generalize the ordinal epsilon numbers—the fixed points of the map \alpha \mapsto \omega^\alpha in ordinal exponentiation—to the surreal setting by defining a surreal \varepsilon as an epsilon number if it satisfies \varepsilon = \omega^\varepsilon, where exponentiation follows the surreal arithmetic defined by ; the positive surreal epsilons coincide with the class of ordinal epsilons. This definition naturally accommodates negative and fractional indices, such as \varepsilon_{-1} = -\varepsilon_0 and \varepsilon_{1/2} constructed as the mediant of \varepsilon_0 and \varepsilon_1. The surreal epsilon function \varepsilon_\alpha enumerates these fixed points via transfinite recursion over ordinal indices \alpha, mirroring the ordinal case but within the full surreal class, yielding increasingly larger fixed points. Formally, in the birthday construction, \varepsilon_\alpha = \left\{ \varepsilon_\beta \cdot \omega^\gamma \;\middle|\; \beta < \alpha,\ \gamma < \omega \right\} \;\middle|\; \emptyset, where the lower set comprises products of prior epsilons with finite powers of \omega, ensuring \varepsilon_\alpha is the simplest surreal greater than all such terms and satisfying the fixed-point equation. Surreal epsilon numbers constitute an "irreducible" subclass of the surreals, closed under addition, multiplication, and other operations in a manner that preserves their fixed-point structure relative to base-\omega exponentiation.

Properties and examples

Surreal epsilon numbers \varepsilon_\alpha, defined for every surreal number \alpha, are fixed points of the surreal exponential function, satisfying \omega^{\varepsilon_\alpha} = \varepsilon_\alpha, where \omega is the first infinite surreal number. These numbers extend the ordinal epsilon numbers into the broader structure of the surreal field, preserving the order such that \varepsilon_\alpha > 0 whenever \alpha > 0. In particular, the surreal \varepsilon_0 coincides exactly with the first ordinal epsilon number, the least fixed point of the ordinal exponential \alpha \mapsto \omega^\alpha. The of all surreal numbers forms a proper larger than the class of ordinals, as there exists a distinct \varepsilon_\alpha for each surreal index \alpha. Their representations in Conway normal form lack components, rendering them infinitesimal-free; moreover, certain subclasses of surreal numbers up to heights below specific numbers are archimedean real-closed fields. Unlike the ordinal numbers, which are closed under ordinal and only in restricted senses due to their well-ordered nature, the set of surreal numbers exhibits closure under surreal and only in limited subclasses. Concrete examples illustrate these distinctions. For negative indices, \varepsilon_{-1} = -\varepsilon_0, the negative counterpart to the first epsilon number, highlighting the of the absent in ordinals. For fractional indices, \varepsilon_{1/\omega} serves as a fixed point \omega^{\varepsilon_{1/\omega}} = \varepsilon_{1/\omega} positioned between \varepsilon_0 and \varepsilon_1, demonstrating the density and continuity of the surreal ordering in the transfinite regime. These numbers play a key role in surreal analysis, particularly in modeling transfinite games within Conway's , where they facilitate the valuation of positions with infinite or advantages.