How did Alfred Tarski show that truth cannot be defined within the same formal language?

Alfred Tarski demonstrated that the concept of truth cannot be defined within the same formal language it describes by highlighting the inherent risk of self-referential paradoxes, specifically the liar paradox.12

If a formal system were powerful enough to define its own truth predicate, one could construct a sentence within that system equivalent to "This sentence is not true". Such a sentence creates a logical contradiction: if it is true, it must be false, and if it is false, it must be true.2

To resolve this, Tarski established that a definition of truth must be constructed in a richer metalanguage—a language used to analyze the object language for which truth is being defined. By separating the two, Tarski was able to define truth compositionally through truth conditions and the notion of satisfaction, without the system needing to contain its own semantic terms. This distinction ensures that the definition of truth remains consistent and free from the contradictions that arise when a language attempts to refer to its own truth-value.1342