https://github.com/ajcr/transfinite

Transfinite ordinal arithmetic and factorisation up to epsilon-zero
https://github.com/ajcr/transfinite

arithmetic factorization ordinals python set-theory transfinite

Last synced: 8 months ago
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Transfinite ordinal arithmetic and factorisation up to epsilon-zero

• Host: GitHub
• URL: https://github.com/ajcr/transfinite
• Owner: ajcr
• License: mit
• Created: 2015-10-07T21:36:07.000Z (over 10 years ago)
• Default Branch: master
• Last Pushed: 2022-03-22T22:00:09.000Z (almost 4 years ago)
• Last Synced: 2025-05-28T04:06:27.425Z (9 months ago)
• Topics: arithmetic, factorization, ordinals, python, set-theory, transfinite
• Language: Python
• Homepage:
• Size: 449 KB
• Stars: 14
• Watchers: 4
• Forks: 1
• Open Issues: 1
• Metadata Files:
  • Readme: README.md
  • Changelog: changelog.md
  • License: LICENSE

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README

          
# transfinite

[![PyPI version](https://badge.fury.io/py/transfinite.svg)](https://badge.fury.io/py/transfinite)

Transfinite [ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) and factorisation up to the first [epsilon number](https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)).

## Installation

Works with Python 3. Can be installed via pip using:

```

pip install transfinite

```

## Usage

For a basic introduction to ordinal arithmetic, look at Wikipedia or see the notebook [here](https://github.com/ajcr/transfinite/blob/master/notebooks/ordinal_arithmetic_basics.ipynb).

Here's a quick demonstration of the library in Jupyter's qtconsole (note that the variable `w` is the first transfinite number). First, some ordinal arithmetic:

![alt tag](https://github.com/ajcr/transfinite/blob/master/images/transfinite_demo.png)

The Ordinal class implements several methods which can be used to check properties of the ordinal:

- `Ordinal.is_limit()`, returns True if the ordinal is a [limit ordinal](https://en.wikipedia.org/wiki/Limit_ordinal).

- `Ordinal.is_successor()`, returns True if the ordinal is a [successor ordinal](https://en.wikipedia.org/wiki/Successor_ordinal).

- `Ordinal.is_gamma()`, returns True if the ordinal is [additively indecomposable](https://en.wikipedia.org/wiki/Additively_indecomposable_ordinal).

- `Ordinal.is_delta()`, returns True if the ordinal is [multiplicatively indecomposable](https://en.wikipedia.org/wiki/Additively_indecomposable_ordinal#Multiplicatively_indecomposable).

- `Ordinal.is_prime()`, returns True if the ordinal is prime.

[Ordinal factorisation](https://en.wikipedia.org/wiki/Ordinal_arithmetic#Factorization_into_primes) into prime ordinals is also implemented. Any composite ordinal `a` can be written as a product of two or more prime ordinals less than `a`:

![alt tag](https://github.com/ajcr/transfinite/blob/master/images/transfinite_demo_2.png)

Note that finite ordinals are not factorised using this method.