Metamath

Metamath is a formal language and a tool for proving mathematical theorems. It is based on a simple, unified syntax that allows for the definition of mathematical terms, axioms, and theorems. The core idea of Metamath is to represent mathematical statements and their proofs as a database of logical statements. This database can then be checked by a Metamath verifier, which ensures the logical consistency of the entire system.

The Metamath language uses a set of simple, unambiguous rules for constructing statements. These rules are themselves formal and can be expressed within the Metamath system, leading to a self-contained and verifiable framework. A key feature is the concept of "proof terms," which are explicit and constructive representations of how a theorem is derived from axioms or previously proven theorems. This explicitness allows for a high degree of confidence in the correctness of proofs.

Metamath has been used to formalize a significant portion of mathematics, including foundational systems like ZFC set theory. Its applications extend to verifying the correctness of software and hardware designs, as well as other areas requiring rigorous logical deduction. The system's emphasis on formal verification makes it a powerful tool for ensuring the soundness of mathematical arguments and complex logical systems.