# Factorial

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The factorial of a natural number n is defined by 0!=1 and a successor recurrence, and equals the product of the positive integers from 1 through n.

mko-factorial · definition · v0.10.0



## Mathematical statement

The function F:N→N is uniquely determined recursively by F(0)=1 and F(n+1)=(n+1)F(n). Its customary notation is n!.





## Representative formula (exact TeX)

F(n)





## Assumptions


- n is a natural number, with natural numbers including 0.





## Symbol scope


- F — representative factorial function
- n — natural-number input
- ! — customary postfix factorial notation





## Intuition, definition, and misconceptions

The value 0! is defined as 1. For a natural number n, (n+1)!=(n+1)·n!, so when n>0, n!=1·2·…·n.

Factorial grows rapidly. Exact computation should use arbitrary-precision integers rather than presenting a floating-point approximation as an exact value.

Factorial occurs in permutation counting, but “n! counts the permutations of an n-element set” is a separate theorem requiring proof; it is not the recursive definition itself.

Common mistakes include treating 0! as 0, reading n! as n multiplied by itself, or presenting an unsafe floating-point value as an exact result.





## Adopted derivation guide

[Factorial: uniqueness of the recursive definition, the product form and computing 5!](https://ocme.evemisslab.com/en/derivations/derivation-mko-factorial-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — Factorial inputs and outputs are natural numbers.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — The successor recurrence and finite product use natural-number multiplication.





## Verification boundary

BigInt can recompute factorial values and recurrence witnesses for finite natural inputs, but does not prove permutation-counting, divisibility, or asymptotic theorems.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind Nat.factorial, factorial_zero, factorial_succ, and this MKO’s natural-number convention.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/factorial.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-factorial/index.md)