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In logic books you read about the truth definition, as an example for what I think is a standard truth definition for FOL: https://en.wikipedia.org/wiki/First-order_logic#Evaluation_of_truth_values.

Now, I also read that Tarski has proven that you cannot define truth.

  1. Am I guessing correctly that Tarski had only proven that a certain kind of truth definition is impossible, so that the standard definition from above does not fall into it?
  2. How can we be sure that our standard definition of truth is consistent and does not bear some hidden contradiction, i.e. can we prove that the standard truth definition is correct/consistent and does not have the defect that lead Tarski to his conclusion?
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If I understand correctly your question, I suspect that you are mistaking two notions of definition.

Tarski's 'definition of truth' is a definition of what means that a first-order structure $M$ satisfies a first-order statement $\sigma$. You can formulate it in $\textsf{ZFC}$ as you can formulate the defintion of group or other mathematical objects or properties.

Tarski's Undefinability Theorem has to do with a technical notion of defintion. A subset $A$ of a first-order structure $M$ (in a suitable language) is definable if there is a formula $\phi(x)$ of the language such that $M \models \phi(a)$ iff $a \in A$. Let $M$ be $\langle \mathbb{N}, +, \cdot\rangle$. Tarski's Theorem says that there is no formula $\phi(x)$ that defines the theory of $M$ ($\text{Th}(M)$, all the sentences true in $M$). Of course $\text{Th}(M)$ is not a subset of $\mathbb{N}$, what is not definable is the set of codes (natural numbers assigned to formulas) in a suitable 'arithmetization of syntax'.

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