Veblen Function

+The Veblen function defines a profound hierarchy of increasingly large [Ordinal Numbers](/wiki/Ordinal_Number), extending the reach of basic [Ordinal Arithmetic](/wiki/Ordinal_Arithmetic) far beyond what simple exponentiation can achieve. It generalizes the concept of [Cantor Normal Form](/wiki/Cantor_Normal_Form), which expresses ordinals using powers of `ω`, and the [epsilon numbers](/wiki/Epsilon_Number), which are the fixed points of `α ↦ ω^α`. The Veblen function achieves this generalization by systematically constructing sequences of [Normal Functions](/wiki/Normal_Function). At its core, each successive function in the hierarchy is defined as the [Fixed Points](/wiki/Fixed_Point) 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](/wiki/Ordinal_Number). This two-argument function generates a remarkably complex and rich structure of ordinals. The hierarchy is defined recursively:

+- `φ_0(α)` is typically defined as `ω^α`. In this context, the epsilon numbers are precisely the fixed points of `α ↦ ω^α`, meaning `ε_α = φ_0(α)` for `α > 0`.

+- For `β > 0`, `φ_β(α)` is defined as the `α`-th common fixed point of all functions `φ_γ` for `γ < β`. More simply, `φ_β(α)` enumerates the fixed points of `φ_ζ` for `ζ < β`.

+- Specifically, `φ_1(α)` enumerates the fixed points of `φ_0(ξ) = ω^ξ`. So, `φ_1(0)` is the first fixed point of `φ_0`, which is the smallest epsilon number, `ε_0`.

+The Veblen function defines a hierarchy of increasingly large [Ordinal Numbers](/wiki/Ordinal_Number), extending basic ordinal arithmetic into vast realms. It generalizes the concept of [Cantor Normal Form](/wiki/Cantor_Normal_Form) and the [epsilon numbers](/wiki/Epsilon_Number) by systematically constructing sequences of normal functions. Each successive function in the hierarchy is defined as the [Fixed Points](/wiki/Fixed_Point) of the previous one, charting the path to incredibly large countable ordinals that are otherwise difficult to name or define.

+The Veblen hierarchy is typically denoted by `φ(α, β)` (or `φ_β(α)`), where `α` and `β` are ordinals. This two-argument function generates a complex structure of ordinals. For instance, `φ(0, α)` corresponds to the epsilon numbers, while `φ(1, 0)` is the [Feferman-Schütte ordinal](/wiki/Feferman-Schütte_Ordinal). 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. It is a cornerstone in the study of [Large Ordinals](/wiki/Large_Ordinals) and [Proof Theory](/wiki/Proof_Theory) as a means to measure the strength of formal systems.

+The Veblen function defines a hierarchy of increasingly large [Ordinal Numbers](/wiki/Ordinal_Number), extending basic ordinal arithmetic into vast realms. It constructs sequences of normal functions, each defined as the [Fixed Points](/wiki/Fixed_Point) of the previous one, charting the path to incredibly large countable ordinals.

+## See also

+- [Ordinal Arithmetic](/wiki/Ordinal_Arithmetic)

+- [Large Ordinals](/wiki/Large_Ordinals)

+- [Recursion Theory](/wiki/Recursion_Theory)