Tarski hierarchies

The general notions of object- and metalanguage are discussed and as a special case of this relation an arbitrary first order language with an infinite model is expanded by a predicate symbol T₀ which is interpreted as truth predicate for. Then the expanded language is again augmented by a new truth predicate T₁ for the whole language plus T₀. This process is iterated into the transfinite to obtain the Tarskian hierarchy of languages. It is shown that there are natural points for stopping this process. The sets which become definable in suitable hierarchies are investigated, so that the relevance of the Tarskian hierarchy to some subjects of philosophy of mathematics are clarified.It should be noticed that these terms object language and meta language have only a relative sense. If, for instance, we become interested in the notion of truth applying to sentences, not of our original object-language, but of our meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language — so to speak, to a meta-language of a higher level. In this way we arrive at a whole hierarchy of languages.(Tarski, 1986, p. 674f)