We aren’t talking about a billion, or a googol (10^100), or even a googolplex (10^(10^100)). Those numbers, while vast, are still within the realm of "finite" in name only. We are talking about numbers so large that the observable universe lacks the atomic real estate to write down their digits.
Building a calculator for this hierarchy requires bridging the gap between standard arithmetic and ordinal arithmetic.
Calculators use "Tree Data Structures" to represent these ordinals. 2. Reduction Rules When a user inputs , the calculator follows a recursive "unwinding" process: is a successor, it expands into a chain of function calls. is a limit, it selects the -th term of that ordinal's fundamental sequence. 3. Approximation Tools
, it is mathematically more powerful than almost anything encountered in standard calculus or physics. To help you dive deeper into specific growth rates: Do you need a between FGH and Hardy hierarchies? Should I explain specific ordinals like ζ0zeta sub 0 or the Feferman-Schütte ordinal?
Instead, an FGH calculator is best implemented as a . It takes a function definition and an input, and it applies the recursive rules until the expression is simplified or evaluated.