Calculate a Finite Hierarchy Level
Choose a finite level and input value. The calculator evaluates the function when the result fits within its integer and operation limits.
Level 2
Repeated application of the previous level. This produces quadratic growth.
Growth category: Quadratic growth
The Finite Fast-Growing Hierarchy
The hierarchy is a sequence of functions indexed by ordinal numbers. At finite levels, each new function is defined by repeatedly applying the previous function.
Fk+1(n) = Fk(n+1)(n)
The notation Fk(m) means applying Fk a total of m times. For example:
F2(n) = 2n+1(n + 1) - 1
The second displayed identity is specific to this page's convention and follows from the chosen iteration rule.
Explore Finite Levels
Level 0
The successor function. It increases an input by one.
Example: F₀(5) = 6
Level 1
Repeated application of the successor function. Under this convention, F₁(n) = 2n + 1.
Example: F₁(5) = 11
Level 2
Repeated application of the previous level. This produces quadratic growth.
Example: F₂(2) = 13
Level 3
A level with polynomial growth of increasing degree as the input grows.
Example: F₃(2) = 65533
Level 4
Repeated iteration of a rapidly growing polynomial-level function.
Example: Values grow very quickly.
Level 5
A further iteration level. Even small inputs can produce values too large for ordinary integer storage.
Example: Exact values may exceed machine limits.
Level 6
The next level in the finite hierarchy, defined by iterating the previous function.
Example: Usually represented symbolically.
Level 7
An increasingly fast-growing finite level.
Example: Usually represented symbolically.
Level 8
Another finite level, constructed by iterating the preceding function.
Example: Usually represented symbolically.
Level 9
A finite hierarchy level with growth far beyond the lower levels.
Example: Usually represented symbolically.
Level 10
A higher finite level in the hierarchy.
Example: Usually represented symbolically.
Level 11
A higher finite level defined by iteration of the previous function.
Example: Usually represented symbolically.
Level 12
A finite level that illustrates the rapid increase in growth caused by hierarchy iteration.
Example: Usually represented symbolically.
Explore Limit Ordinal Levels
The hierarchy can also be extended beyond finite indices. Limit ordinals require fundamental sequences: sequences of smaller ordinals that approach the limit ordinal.
What Is the Fast-Growing Hierarchy?
The fast-growing hierarchy is a family of functions used in mathematical logic, proof theory and the study of computational complexity. It organizes functions according to increasingly powerful growth rates.
Finite ordinal levels
At finite indices, the hierarchy is generated by iteration. Each level repeatedly applies the preceding level, producing faster growth as the index increases.
The first limit ordinal: omega
The ordinal ω is the first infinite ordinal. To define a hierarchy function at ω, a fundamental sequence is used. In this calculator, the selected convention is:
Beyond omega
Higher levels such as ω + 1, ω · 2, ω² and ε₀ require more elaborate ordinal notation and fundamental-sequence choices. The exact hierarchy depends on the selected formal system.
Fast-Growing Hierarchy and Other Large-Number Systems
The hierarchy is related to other fast-growing mathematical constructions, but the systems are not interchangeable.
| System | Main idea | Typical use |
|---|---|---|
| Exponentiation | Repeated multiplication | Elementary arithmetic |
| Tetration | Repeated exponentiation | Hyperoperations |
| Pentation | Repeated tetration | Higher hyperoperations |
| Fast-growing hierarchy | Ordinal-indexed function iteration | Proof theory and logic |
| Busy Beaver function | Maximum behavior of halting machines | Computability theory |
| Hardy hierarchy | Ordinal-indexed functions | Ordinal analysis |
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Frequently Asked Questions
What is the fast-growing hierarchy?
It is a sequence of increasingly fast-growing functions indexed by ordinal numbers. It is used in proof theory, mathematical logic and the study of very large growth rates.
What does F₀(n) represent?
In this calculator's convention, F₀(n) = n + 1. It is the successor function and forms the starting point of the finite hierarchy.
What is Fω(n)?
It is the first limit-level function. Under the fundamental-sequence convention used here, Fω(n) = Fₙ(n).
What is epsilon zero?
ε₀ is the smallest ordinal satisfying ω^ε₀ = ε₀. It is an important ordinal in proof theory and ordinal analysis.
Can this calculator evaluate every hierarchy level?
No. It evaluates selected finite levels when the result fits within PHP's integer range and operation budget. Higher levels are shown symbolically because their values can exceed ordinary computing limits extremely quickly.
Does a higher hierarchy level always grow faster?
For the standard finite hierarchy used here, higher levels eventually dominate lower levels. Comparisons involving limit ordinals require the same fundamental-sequence and hierarchy conventions to be specified.
Is the fast-growing hierarchy the same as hyperoperations?
No. Hyperoperations extend arithmetic through operations such as exponentiation and tetration. The fast-growing hierarchy is based on iterating functions indexed by ordinals. Both describe rapid growth, but their definitions differ.