The Fast-Growing Hierarchy (FGH) is a family of functions used in mathematics and computer science to classify the growth rates of functions. It is the gold standard for measuring the size of large numbers, from the merely huge (like $10^100$) to the incomprehensibly large (like Graham’s Number and TREE(3)).
is where standard calculators break down completely. Because is a limit ordinal, dynamically evaluates to (which is the multi-million-digit number mentioned above). Calculation:
This famously massive number is bounded tightly between in extended versions of the hierarchy, where represents the first transfinite ordinal. How an FGH Calculator Operates fast growing hierarchy calculator
Because the actual numbers cannot be stored in standard computer memory, the calculator outputs the number using alternative notations (e.g., Knuth's Up-Arrow, Conway Chained Arrow, or Bowers Exploding Array Notation). The Challenge of Computability
if alpha == 0: return f"prefix = n+1"
This script acts as a symbolic calculator. It can compute values for $f_0, f_1, f_2, \dots, f_\omega$. Note that $f_3(3)$ already yields a number with over 3 trillion digits. This program will stop if the number becomes too large to store in memory, but it will print the reduction steps for any valid input.
is an ordinal number. Its recursive definition is remarkably simple, yet it leads to explosive growth: The Fast-Growing Hierarchy (FGH) is a family of
, you can often calculate or approximate values manually using these standard shortcuts: Code Golf Stack Exchange (Successor) (Doubling) (Exponential growth) (Tetration/Tower growth) Technical Implementations
that supports both FGH and SGH (Slow-Growing Hierarchy) calculations up to Rathjen's capital Quick Reference for Lower Levels For levels below Because is a limit ordinal, dynamically evaluates to