---
license: cc0-1.0
tags:
  - theorem-proving
  - formal-methods
  - metamath
  - logic
  - set-theory
  - mathematics
task_categories:
  - text-generation
  - feature-extraction
language:
  - en
size_categories:
  - 10K<n<100K
---

# Metamath

A structured dataset of formally verified theorems and axioms from **Metamath**, one of the largest collections of rigorously verified mathematics in the world.

## Source

- **Repository:** https://github.com/metamath/set.mm
- **Website:** https://us.metamath.org
- **License:** CC0 1.0 (Public Domain)

## Statistics

| Property | Value |
|----------|-------|
| **Total Entries** | 72,962 |
| **Theorems** | 69,394 |
| **Axioms** | 3,568 |
| **Docstring Coverage** | 95.6% |

### Database Distribution

| Database | Entries | Description |
|----------|---------|-------------|
| set.mm | 49,410 | Classical logic and ZFC set theory |
| iset.mm | 15,953 | Intuitionistic logic and set theory |
| nf.mm | 6,245 | New Foundations set theory |
| ql.mm | 1,176 | Quantum logic |
| hol.mm | 178 | Higher-order logic |

## Schema

| Column | Type | Description |
|--------|------|-------------|
| `fact` | string | Statement in form "label : assertion" |
| `type` | string | "theorem" or "axiom" |
| `library` | string | Source database (set, iset, nf, ql, hol) |
| `imports` | list[string] | Empty (Metamath uses single-file structure) |
| `filename` | string | Source .mm file |
| `symbolic_name` | string | Theorem/axiom label |
| `docstring` | string | Description with contributor attribution |

## Metamath Syntax

Metamath uses a minimal syntax:
- `|-` indicates a provable statement
- `/\` is conjunction (AND)
- `\/` is disjunction (OR)
- `->` is implication
- `<->` is biconditional
- `-.` is negation
- `A.` is universal quantifier
- `E.` is existential quantifier

## Notable Theorems

The set.mm database includes proofs of major results:
- Fundamental Theorem of Calculus
- Prime Number Theorem
- Cauchy-Riemann equations
- Hahn-Banach theorem
- And thousands more...

## Creator

Charles Norton ([phanerozoic](https://huggingface.co/phanerozoic))