This is a collection of rigorously verified Metamath databases that specify mathematical axioms and formal proofs of theorems derived from those axioms.
Metamath is a computer language and associated computer program for archiving, verifying, and studying mathematical proofs.
Unlike some other systems, Metamath does not build a particular set of axioms into the system. Instead, the Metamath language is simple and robust, with an almost total absence of hard-wired syntax. In Metamath you express the axioms, theorems, and their proofs in a database (set of text files). To prove a theorem, every proof step must be proven using an axiom or a previously proven theorem; as a result, nothing is hidden. We believe Metamath provides about the simplest possible framework that allows essentially all of mathematics to be expressed with absolute rigor.
The resulting databases provide human-readable axioms and theorem statements. We compress proofs in this repository, so their proofs take relatively little space, but tools can easily decompress them to provide a human-readable sequence of every proof step. Metamath verification is incredibly fast; the largest database available can be re-verified within seconds by some verifiers.
For more information see the Metamath Home Page, the Metamath Proof Explorer Home Page, the Metamath book, or the video "Metamath Proof Explorer: A Modern Principia Mathematica".
The databases included and links to their generated displays, in (approximate) decreasing size, are:
- "set.mm" aka "Metamath Proof Explorer (MPE)" - uses classical logic and Zermelo–Fraenkel set theory with the axiom of choice (ZFC). This is one of the largest collections of formally verified mathematics in the world (e.g., it has completed many challenge theorems in the Formalizing-100 Theorems challenge list); this video visualizes set.mm's growth through 2020-04-29. [Generated display]
- "iset.mm" aka "Intuitionistic Logic Explorer" - uses intuitionistic set theory. In particular, it does not presume that the law of excluded middle is necessarily true in all cases. [Generated display]
- "nf.mm" aka "New Foundations Explorer" - constructs mathematics using Quine's New Foundations (NF) set theory axioms, a direct derivative of the "hierarchy of types" set theory originally presented in Whitehead and Russell's Principia Mathematica. [Generated display]
- "ql.mm" aka "Quantum Logic Explorer" - Starts from the orthomodular lattice properties proved in the Hilbert Space Explorer and takes you into quantum logic. [Generated display]
- "hol.mm" aka "Higher-Order Logic Explorer" - Starts with higher-order logic (HOL, also called simple type theory) and derives equivalents to ZFC axioms, connecting the two approaches. [Generated display]
- "peano.mm" - Peano arithmetic.
- "big-unifier.mm" - a unification and substitution test for Metamath verifiers, where small input expressions blow up to thousands of symbols. It is a translation of William McCune's "big-unifier.in" for the OTTER theorem prover.
- "miu.mm" - a simple formal system for use as a demonstration based on work by Hofstadter.
- "demo0.mm" - a simple formal system used as a demonstration in Chapter 2 of the Metamath book.
Feel free to also look at this list of other Metamath databases.
We work to provide extremely high confidence that the proofs are completely correct in these databases, especially for the set.mm and iset.mm databases (the primary databases under active development).
Changes ("commits") to any database are first automatically verified before they are accepted, using GitHub actions. Every change to set.mm and iset.mm is verified by many independent verifiers, including metamath-exe, which also checks markup, and mmj2, which checks definition soundness. All other databases' proofs are verified by one verifier (metamath-exe) in every commit. For more information, see VERIFIERS.md.
We're a friendly community, and we would love to have more collaborators!
See CONTRIBUTING.md for more information on how to create a contribution, as well as how contributions are evaluated to decide if they will be merged in. That file also has some ways to contact us if you'd like help contributing.
Our sincere thanks to everyone who has contributed to this work.
This entire collection is dedicated to the memory of Norman "Norm" Dwight Megill, Ph.D. (1950-2021), creator of the Metamath system and cultivator of an international community of people with the shared dream of digitizing and formally verifying mathematics. His ideas and design have been influential in formal mathematics around the world. He is missed.