Generate von Neumann Ordinals

The Generate von Neumann Ordinals tool lets you create a list of von Neumann-style ordinals up to any non-negative number you type in. These are sets where each number is represented by the set of all smaller numbers so 0 is [], 1 is [[]], 2 is [[], [[]]], and it keeps building. It’s a neat way to visualize set-theoretic counting, and this tool makes that super simple.

You can tweak how the output looks with pretty formatting or include the nesting depth for each ordinal. Great if you’re learning set theory or just want to see what ordinals look like in JSON form.

Total integers: 0
Options
Allow duplicates
Sort output
Maximize output
Output format:

How to Use:

  • Enter a number in the input box this is the highest ordinal you want to generate.
  • The output updates automatically, showing all ordinals from 0 up to your number.
  • Use the “Pretty format” toggle to get nicely spaced JSON.
  • Use the “Show nesting depth” toggle to include the depth value alongside each ordinal.
  • Use the “Maximize output” toggle if you want to expand the output box for better visibility.
  • Click Copy Output to copy the JSON.
  • Click Export to File to download it as a .json file.
  • Click Clear All to reset everything.

What Generate von Neumann Ordinals can do:

This tool’s perfect for anyone studying set theory, building logic-related tools, or just curious about how ordinals grow. You can quickly visualize how von Neumann ordinals build on each other and flip between raw and formatted views. Showing the depth is handy when you want both structure and value, and the live output means you don’t have to keep hitting buttons to see changes. It’s all reactive.

Example:

Input:

3

Output (Pretty format ON, Show depth OFF):

[
  [],
  [[]],
  [[], [[]]],
  [[], [[]], [[], [[]]]]
]

Output (Pretty format ON, Show depth ON):

[
  { "ordinal": [], "depth": 0 },
  { "ordinal": [[]], "depth": 1 },
  { "ordinal": [[], [[]]], "depth": 2 },
  { "ordinal": [[], [[]], [[], [[]]]], "depth": 3 }
]

Common Use Cases:

If you’re learning how ordinals work in foundational mathematics, this makes abstract ideas visual and concrete. You can also use the tool to generate mock data for logic simulators or educational content. It’s also useful in classrooms or textbooks to demo set-based construction without having to write it all manually.

Useful Tools & Suggestions:

If you’re exploring sets with Generate von Neumann Ordinals, try Generate Zermelo Ordinals too it’s another way of defining numbers using set theory. And if you want to see the raw values involved, Enumerate Integers helps visualize each step.

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