). It requires specific formatting, such as writing "p" for the symbol Hardy Hierarchy Calculator : This tool uses the ExpantaNum.js library to handle transfinite ordinals like omega raised to the omega power
For many, the quickest way to interact with FGH is through dedicated online calculators.
Users input strings like f_(omega^2 + omega*2 + 1)(4) . The calculator must parse this into an abstract syntax tree (AST) that natively represents ordinals. It validates whether the ordinal is well-formed and determines its type (zero, successor, or limit). The Expansion Engine
: Showing the step-by-step expansion of fundamental sequences.
Enter the . However, not all calculators are created equal. Most are buggy, limited to low ordinals, or fail to handle fundamental sequences correctly. This article explores what makes a high-quality FGH calculator, why you need one, and how to separate the gold from the pyrite in the world of ordinal analysis. fast growing hierarchy calculator high quality
(the limit of Peano Arithmetic) and the Feferman-Schütte ordinal Γ0cap gamma sub 0 How to Verify the Quality of an FGH Calculator
: An "Ordinal Expander" written in JavaScript. It's designed to compute the fundamental sequences for a modified version of the Extended Buchholz function. This tool is highly regarded within the googology community for its utility in investigating OCF-based ordinal notations.
Ensure all ordinals are in Cantor normal form or Veblen normal form to compare equality.
A calculator for FGH must handle:
that can display fundamental sequences and calculate both FGH and SGH (Slow-Growing Hierarchy) up to high ordinals like Rathjen's Quick Reference: How FGH Grows
: These are two gold-standard web-based calculators designed by googologist Denis Maksudov:
Compute λ[n] on demand, cache results for repeated indices.
Standard calculators stop at integers. A high-quality tool supports: (Omega): The first infinite ordinal. ϵ0epsilon sub 0 (Epsilon-zero): The limit of the sequence , used to reach the Feferman-Schütte ordinal ( Γ0cap gamma sub 0 2. Implementation of Fundamental Sequences To calculate The calculator must parse this into an abstract
) to "diagonalize" and move beyond finite numbers into the realm of ϵ0epsilon sub 0 , and beyond. What Makes a "High-Quality" FGH Calculator?
The community standard for testing large number functions.
The development of a "fast-growing hierarchy calculator" represents a significant advancement in the ability to compute and understand these rapidly growing functions. A high-quality calculator for this purpose would not only compute the values of functions within the hierarchy but also provide insights into their growth rates, perhaps even visualizing how quickly these functions expand.