Transfinite Ordinal Calculator

Enter Ordinal Expression:

❓ Real number representation of the ordinal. ↔️

Graphical Representation:

Result will appear here.

Linear String Representation (CNF):

Result will appear here.

How to Use

This tool parses expressions representing ordinal numbers and calculates their Cantor Normal Form.

Use non-negative integers (e.g., 0, 1, 5, 123).

Use w to represent the smallest infinite ordinal, omega (ω).

Use e_0 to represent epsilon-naught (ε₀), the first fixed point of ω^x=x.

Supported operations: + (addition), * (multiplication), ^ (exponentiation), ^^ (tetration).

Use parentheses () for grouping.

Standard operator precedence is followed: ^^ (highest, right-associative), then ^ (right-associative), then * (left-associative), then + (left-associative).

Examples:

1+w ➜ w

w+1 ➜ w+1

(w+1)*2 ➜ w*2+1

w*w ➜ w^2

2^w ➜ w

w^(w+1) ➜ w^(w+1)

(w^w)^w ➜ w^(w^2)

2^^3 (2↑↑3) ➜ 16 (finite repeated exponentiation: 2^(2^2) = 2^4 = 16)

w^^2 (ω↑↑2) ➜ w^w

w^^w (ω↑↑ω) ➜ e_0

2^^w (2↑↑ω) ➜ w

e_0+1 ➜ Error(Currently unsupported)

w+e_0 ➜ e_0

About This Calculator

This Ordinal Number Calculator is a tool designed to explore the fascinating world of transfinite arithmetic. It takes ordinal expressions written with non-negative integers, the symbol w (omega), standard arithmetic operators (+, *, ^, ^^), and parentheses, and computes their Cantor Normal Form (CNF) or evaluates them according to defined rules up to ε₀ (epsilon-naught).

The Cantor Normal Form provides a unique representation for ordinals less than epsilon-naught (ε₀). This calculator implements the standard rules for ordinal addition, multiplication, exponentiation, and tetration to ensure accurate CNF results.

Key Features:

Parses complex ordinal expressions, including those with tetration (^^) and epsilon-naught (e_0).

Calculates accurate Cantor Normal Form based on established ordinal arithmetic rules.

Displays results in both a traditional linear string format and a graphical representation (e.g., using ω, ε₀, and ω↑↑n for ω tetrated to n).

Automatically simplifies very complex results based on a structural complexity budget to enhance readability. If simplification occurs, an indicator (e.g., (Displayed complexity: G_simp / G_orig)) is shown.

Provides an ordinal-to-real number mapping `f(α)` for the original, unsimplified ordinal. This mapping is parameterized and its range (e.g., for the slider) is set dynamically.

Allows copying of both the text and graphical (as image) representations.

Includes an operation budget to prevent excessively long computations for extremely complex inputs.

A particularly useful feature is the interactive ordinal explorer. Use the slider (located just below the expression input area) to select a real number (the range of the slider is dynamically set based on the current `f(α)` mapping parameters). The calculator will then instantly display the transfinite ordinal α that corresponds to this real number (where f(α) ≈ your selected value), and it will also update the main "Enter Ordinal Expression:" box with this ordinal's string representation. The ↔️ (nudge) control next to the slider allows for fine-grained adjustments. This provides a powerful way to discover and understand different ordinal structures by exploring their f(α) mappings.

This project was developed to provide an accessible way to experiment with and visualize ordinal numbers up to the first epsilon number.

Developed by Gemini 2.5 Pro, with the guidance of ordinal number enthusiast Meni Rosenfeld. You can read more about ordinal numbers here: The Unabashed Expanse of Ordinal numbers.

Calculation results were compared against Claudio Kressibucher's Ordinal Calculator to verify accuracy. There might still be mistakes - if you find any, please let me know!