  Although some work has been done on the metamathematics of Metamath, there
has not been a clear definition of a model for a Metamath formal system. We
define the collection of models of an arbitrary Metamath formal system, both
for tree-based and string-based representations. This definition is
demonstrated with examples for propositional calculus, $\textsf{ZFC}$ set
theory with classes, and Hofstadter's MIU system, with applications for proving
that statements are not provable, showing consistency of the main Metamath
database (assuming $\textsf{ZFC}$ has a model), developing new independence
proofs, and proving a form of G\"odel's completeness theorem.
