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?)

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    $\begingroup$ I wouldn't necessarily say that Tarski "resolves" the Liar paradox. Rather, I think it's better to view the Liar itself as a very broad theorem - along the lines of "There is no consistent formal system capable of both self-reference and defining truth" (although making this precise takes serious work!). This Liar Theorem isn't something which needs resolution, it's simply true. Tarski's Undefinability Theorem is then a corollary of the Liar Theorem and the Diagonalization Lemma. (In general I think it's helpful to recast "paradoxes" as theorems and then look for concrete corollaries.) $\endgroup$ Aug 25 '21 at 16:40
  • $\begingroup$ @NoahSchweber Can you elaborate on what you mean by "simply true"? The resolution as far as I understand it, does not allow for the truth of the statement to even exist within the hierarchy (more precisely, that it cannot be expressed), and Tarski's theorem is a corollary of the diagonal lemma & self-reference. Saying in $\mathbf{PA}$ something along the lines of: "this sentence is false" would be, according to a Tarskian system, impossible (as written above). Regardless, I still think this doesn't quite help me here as I want to understand, if & why, it "resolves" it, and also its generality. $\endgroup$
    – Math3147
    Aug 25 '21 at 17:31
  • $\begingroup$ I meant that the result saying (something like) "There is no consistent formal system capable of both self-reference and defining truth" is simply true. The Liar paradox is the thing we consider in the course of proving this theorem. $\endgroup$ Aug 25 '21 at 17:44
  • $\begingroup$ @NoahSchweber Oh I see. I was already aware of this but I appreciate your recommended insight. It is a corner stone of the first incompleteness theorem to understand what you said between quotes. Regardless, I'm still baffled as to why it would be true in every system... the Tarskian system seems to probe Peano Arithmetic more so than any system; I could see this being generalized to any system capable of arithmetic as is already done but to any formal system? I could see the intuition but I've not seen the proof at least (I've tried to look it up but no one's discussing it with generality.) $\endgroup$
    – Math3147
    Aug 26 '21 at 11:37
  • $\begingroup$ Edit: A certain level of arithmetic not all arithmetic. Also, in every formal system not just any system (I don't want to include paracomplete systems in here; they already accept those trivially.) $\endgroup$
    – Math3147
    Aug 26 '21 at 11:42

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.

To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

However, this system is incomplete. One would like to be able to make statements such as "For every statement in level α of the hierarchy, there is a statement at level α+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible). Saul Kripke is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth," and it is recognized as a general problem in hierarchical languages.

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.

An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula A to its truth value ||A||, and the "semantic denotation function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows: No sufficiently powerful language is strongly-semantically-self-representational.

To prove the first incompleteness theorem, Gödel represented statements by numbers. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number. It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.

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.

  • $\begingroup$ I understand and already know this. But the generality of it is not clear, and the proof addresses Peano Arithmetic mainly, and also there are paradoxes (check out revenge paradoxes, this gets a bit philosophical unfortunately) with Tarski's truth predicate, and does this mean it is the only predicate? What about mutli-valued logics? Etc. ? $\endgroup$
    – Math3147
    Sep 1 '21 at 17:19
  • $\begingroup$ @Math3147 I don't know if below linked content from same wiki reference helpful to you. "Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value." This is called correspondence theory of truth in philosophy jargon, so Kripke treats Tarski's semantic theory as correspondence, not deflationary... $\endgroup$
    – mohottnad
    Sep 1 '21 at 18:34
  • $\begingroup$ Can you link something about Kriple's comments? Although they're more philosophy related than related to arithmetic or decidability. $\endgroup$
    – Math3147
    Sep 1 '21 at 18:38
  • $\begingroup$ @Math3147 it's in the second link in my answer above (just below Tarski's solution section). IMHO philosophy is all about awareness not some niffty-gritty technical or formal details, and most people need to be aware of something first then dive into details later, then spiral (not cyclic hopefully). $\endgroup$
    – mohottnad
    Sep 1 '21 at 18:41
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    $\begingroup$ I meant a more comprehensive read. Philosophy to me sounds like speculative theories that don't actually prove what their talking about. I'm all for intuition... one that is grounded. Plus, this site mainly discusses mathematics too, and so it is slightly off-topic. $\endgroup$
    – Math3147
    Sep 1 '21 at 18:44

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