We present a generalization of K-theory to operator systems. Motivated by spectral
truncations of noncommutative spaces described by C*-algebras and inspired by
the realization of the K-theory of a C*-algebra as the Witt group of hermitian
forms, we introduce new operator system invariants indexed by the corresponding
matrix size. A direct system is constructed whose direct limit possesses a semigroup
structure, and we define the K0-group as the corresponding Grothendieck group.
This is an invariant of unital operator systems, and, more generally, an invariant
up to Morita equivalence of operator systems. For C*-algebras it reduces to the
usual definition. We illustrate our invariant by means of spectral truncations and
the spectral localizer.