I was looking in the literature, and in my textbook, it was concluding Tarski's theorem after showing: $$\mathbf{PA} \vdash \varphi \; \longleftrightarrow \; \lnot \text{truth}(\ulcorner{\varphi}\urcorner)$$ Then it tells that in order to find a model $\mathbb{N}$ such that it models $\varphi$, we would need to resolve Liar's paradox which is a contradiction.
More formally, Tarski's theorem states that:
\begin{gather*}
\text{There is no}~\mathcal{L}_{PA}\text{-formula truth(}x)~\text{with one free variable}~x~\text{such that}~\mathbb{N} \models \text{truth(}\#\varphi) \leftrightarrow \varphi.
\end{gather*}
By the Diagonalisation Lemma there exists an $\mathcal{L}_{PA}$-sentence $\sigma$ such that: $$\mathbf{PA} \vdash \sigma \; \longleftrightarrow \; \lnot \text{truth}(\ulcorner{\sigma}\urcorner)$$ Also, then: \begin{equation}\mathbb{N} \models \text{truth}(\#\sigma) \; \iff \; \mathbb{N} \models \sigma \; \iff \; \mathbb{N} \models \lnot \text{truth}(\#\sigma) \tag{$\bot$}\end{equation}
This would disallow for a statement such as: "This sentence is false" from existing as a truth in $\mathbf{PA}$. My question now is, does this resolve Liar's paradox truly or are there objections to this? Also, is this the only form of Liar's paradox, and if so, how general is Tarski's solution here (in terms of applicability to others forms of Liar sentences)?
Edit: I've also seen comments on the problem being undecidable, and thus incomplete in terms of knowing the "truth value" of it. If a solution (in a perhaps multi-valued logic) is proposed, then doesn't that counter the fact that it is incomplete or are the results inconsistent? What follows from what exactly? (Also, did Tarski make some helpful comments on it?)
Best Answer
Tarski's famous semantic theory of truth asserts a truth-predicate for the sentences of a given formal language cannot be defined within that language, see here. There're other solutions for Liar.
So Tarski's undefinability theorem is a powerful and philosophically fresh way to look at semantic expressiveness limitation about all conceivable truth predicates of any formal language. Of course Tarski’s theorem was developed in a bivalent system, if you're using another truth theory with multivalued-logic such as fuzzy logic, the same reference mentions it can resolve the liar paradox as having truth-value=0.5. Kripke also circumvented the consequences of Tarski’s theorem by using three-valued logic as referenced here which discussed all your concerns in detail including revenge paradox. For an axiomatic formal theory of truth see SEP article here.