Compare successive levels of a fast-growing hierarchy, calculate selected function values, and discover how repeated iteration transforms relatively simple functions into extraordinarily fast-growing ones.
Explore the hierarchyA fast-growing hierarchy is a family of functions arranged in levels. Each higher level is constructed by iterating the function from the previous level.
Here, Fk(n+1) means applying the function Fk repeatedly n + 1 times, starting with n. This page uses this specific finite-level convention throughout.
The selected function value exceeded the numerical or computational limits of this explorer. Its symbolic representation is:
Iterates the previous exponential-type function. Growth becomes extremely rapid.
Adds one to the input. This is the starting level of the hierarchy.
Repeats the successor function n + 1 times, starting at n.
Repeats the linear function n + 1 times. Its values grow exponentially.
Iterates the previous exponential-type function. Growth becomes extremely rapid.
Iterates the third level. Even small inputs can produce values beyond ordinary numerical representation.
A further iteration of the previous level, illustrating the explosive increase across hierarchy levels.
Calculate the same input at each selected level. This makes the effect of increasing the hierarchy index easier to see.
The hierarchy begins with a simple successor function. Applying that function repeatedly creates linear growth. Repeating the next level creates exponential-type growth. Further levels repeatedly iterate increasingly powerful functions.
Each level is built from the previous one. This is why increasing the hierarchy index can have a much greater effect than merely increasing the input by one.
| Level | Definition | General behavior |
|---|---|---|
| F0 | n + 1 | Successor |
| F1 | 2n + 1 | Linear |
| F2 | 2n+1(n+1) − 1 | Exponential-type |
| F3 | Iterated F2 | Repeated exponential-type growth |
| F4 | Iterated F3 | Much faster growth |
| F5 | Iterated F4 | Higher hierarchy growth |
Hierarchies provide a systematic language for comparing functions that grow at very different rates. They are useful in mathematical logic, combinatorics, computability theory, and the study of extremely large numbers.
Standard computer number types have finite precision and finite range. Higher hierarchy levels can exceed those limits even when their mathematical definitions are compact and exact.
It is a family of functions arranged in levels, where each higher function is defined by iterating the previous function. It provides a way to study extremely rapid growth in mathematics.
In this explorer, F₀(n) = n + 1. It is the successor function and the starting point of the hierarchy.
F₁(n) = 2n + 1 under the iteration convention used here. It is obtained by applying F₀ exactly n + 1 times, starting with n.
F₂(n) = 2n+1(n + 1) − 1. It is obtained by iterating F₁ n + 1 times.
Different authors choose different starting functions, iteration counts, and indexing conventions. The same hierarchy level can therefore represent different functions in different sources. Always check the definitions before comparing numerical values.
No. This explorer covers a finite hierarchy indexed by nonnegative integers. Transfinite versions use ordinal indices and require additional definitions, including fundamental sequences at limit ordinals.
The Ackermann function is closely related to fast-growing hierarchies. Its growth is faster than every primitive recursive function, and particular hierarchy conventions can describe its growth through selected levels.
Some values exceed ordinary floating-point limits or the computation budget. The explorer displays symbolic notation instead of inventing a numerical approximation.