The Veblen function defines a profound hierarchy of increasingly large Ordinal Numbers, extending the reach of basic Ordinal Arithmetic far beyond what simple exponentiation can achieve. It generalizes the concept of Cantor Normal Form, which expresses ordinals using powers of ω, and the epsilon numbers, which are the fixed points of α ↦ ω^α. The Veblen function achieves this generalization by systematically constructing sequences of Normal Functions. At its core, each successive function in the hierarchy is defined as the Fixed Points of the previous one, charting a systematic path to incredibly large countable ordinals that are otherwise difficult to name or define within simpler systems.
The Veblen hierarchy is formally denoted by φ(α, β) (or often φ_β(α) in some contexts), where α and β are Ordinal Numbers. This two-argument function generates a remarkably complex and rich structure of ordinals. The hierarchy is defined recursively:
φ_0(α)is typically defined asω^α. In this context, the epsilon numbers are precisely the fixed points ofα ↦ ω^α, meaningε_α = φ_0(α)forα > 0.- For
β > 0,φ_β(α)is defined as theα-th common fixed point of all functionsφ_γforγ < β. More simply,φ_β(α)enumerates the fixed points ofφ_ζforζ < β. - Specifically,
φ_1(α)enumerates the fixed points ofφ_0(ξ) = ω^ξ. So,φ_1(0)is the first fixed point ofφ_0, which is the smallest epsilon number,ε_0. - The ordinal
φ(1, 0)is commonly associated with the Feferman-Schütte ordinal, often denotedΓ_0, which represents the proof-theoretic strength of certain formal systems of Arithmetic.
The function's ability to "scale" to produce extremely large ordinals is evident when considering terms like φ(n,n) for Natural Numbers n. This represents a diagonal application of the function, yielding ordinals that grow incredibly fast, far exceeding those generated by simply iterating simpler ordinal functions. The Veblen function is a cornerstone in the study of Large Ordinals and Proof Theory, particularly in the field of Ordinal Analysis, as a precise means to measure the consistency strength of formal mathematical systems.
