Full Idea
In semantic theories of truth (Tarski or Kripke), a truth predicate is defined for an object-language. This definition is carried out in a metalanguage, which is typically taken to include set theory or another strong theory or expressive language.
Gist of Idea
In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage
Source
Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1)
Book Reference
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.2
A Reaction
Presumably the metalanguage includes set theory because that connects it with mathematics, and enables it to be formally rigorous. Tarski showed, in his undefinability theorem, that the meta-language must have increased resources.
Related Idea
Idea 15650 Axiomatic theories of truth need a weak logical framework, and not a strong metatheory [Halbach]