The Fast-Growing Hierarchy (FGH) is a system of functions used in googology to name and categorize unimaginably large numbers. It outpaces standard notation like exponents or even Knuth's up-arrows by using transfinite ordinals. Core Functionality The hierarchy, denoted as , builds speed based on the index (the "ordinal") and the input Zero Stage: . This is simple successor logic. Successor Stage: . The function iterates itself Limit Stage: For limit ordinals (like ), we use a fundamental sequence: Notable Benchmarks As the index increases, the growth rate explodes. : Equal to . Linear growth. : Equal to . Exponential growth. : Comparable to Graham’s Number. It uses power towers.
Limit Rule (Limit Ordinals): If $\alpha$ is a limit ordinal (like $\omega$ or $\omega \times 2$), we use fundamental sequences. $$f_\alpha(n) = f_\alpha[n](n)$$ Translation for the calculator: Find the $n$-th element in the fundamental sequence of $\alpha$ and evaluate that function. fast growing hierarchy calculator
is an ordinal number. Its recursive definition is remarkably simple, yet it leads to explosive growth: The Fast-Growing Hierarchy (FGH) is a system of
If you did compute ( f_\omega+1(4) ) as an integer, you’d need more than ( 10^100 ) bits of memory—physically impossible. Hence any honest FGH calculator never expands to a full integer; it stays in a compressed symbolic form unless the result is tiny. User-friendly interface : The calculator has a simple