The Metalanguage Hierarchy and Semantics in Mathematics

Models, Truth, and Metalanguage established that talk of truth and models must ascend from an object language to a strictly richer metalanguage — by Tarski's undefinability theorem, a language cannot define its own truth predicate. This remark follows that idea upward: if every semantic vocabulary needs a metalanguage above it, what stops an infinite regress? And what does this tower mean for how semantics actually functions in mathematical practice?

It builds on the object-theory/metatheory distinction from Set Theory and First-Order Logic.

Tarski's hierarchy of languages

Tarski's response to the Liar paradox was not just a negative theorem but a positive prescription: stratify language into levels.

• $L_0$ — the object language: it talks about a domain of objects (numbers, sets, …) but contains no semantic predicates of its own. It can say "$2+2=4$" but not "'$2+2=4$' is true."
• $L_1$ — the metalanguage for $L_0$: it contains a truth predicate ${\mathrm{True}}_0(┌\phi ┐)$ applying to (codes of) $L_0$-sentences, plus the apparatus to talk about $L_0$'s syntax. Tarski's Convention T fixes its meaning by requiring every instance of the schema $${\mathrm{True}}_0(┌\phi ┐)\leftrightarrow \phi$$ (e.g. ${\mathrm{True}}_0(┌\text{snow is white}┐)\leftrightarrow \text{snow is white}$).
• $L_2$ — a metalanguage for $L_1$: it has a predicate ${\mathrm{True}}_1$ for $L_1$-sentences, including those that already mention ${\mathrm{True}}_0$.
• … and so on: $L_{n+1}$ supplies the truth predicate that $L_n$ provably cannot contain.

Each level can define truth for the level below, never for itself. The Liar — "this sentence is false" — is dissolved because there is no single, level-free predicate "false" for it to invoke: a sentence at level $n$ can only be evaluated by ${\mathrm{True}}_n$ living at level $n+1$, so the self-reference never closes.

Does the regress ever stop?

Three things keep the hierarchy from being a vicious infinite regress.

1. The tower is potential, not actual. One never needs all levels at once. Any particular piece of reasoning uses finitely many: an object language and one metalanguage above whatever semantic talk it requires. The hierarchy is a resource you climb only as far as the task demands.
2. Levels can collapse into a single strong theory. In practice the metatheory is usually just set theory (ZFC), which is rich enough to define a satisfaction predicate $\mathrm{Sat}(\mathcal{M},┌\phi ┐)$ for any set-sized structure $\mathcal{M}$ and any object theory $T$ whose language is a set. ZFC thereby serves as metalanguage for all the usual object theories at once — including weaker fragments of itself. What it still cannot do, on pain of Tarski, is define truth for its own full language; that single missing predicate is what a genuine step up (e.g. ZFC + "there is a truth class," or a large-cardinal extension) would supply.
3. Transfinite iteration is possible but optional. One can index levels by ordinals, $L_{\alpha }$ for $\alpha$ into the transfinite, and study revision theories or Kripke's fixed-point construction. Kripke's alternative abandons the strict hierarchy for a partial truth predicate built as a least fixed point, allowing a language to contain its own (gappy, paracomplete-flavored) truth predicate at the cost of the Liar being undefined rather than banned. This is the principal modern rival to Tarski's stratification.

So the regress is real but benign: potentially infinite, actually finite-on-demand, and usually short-circuited by adopting one sufficiently strong metatheory.

Can a theory be its own metatheory? (German analysing English)

A natural test of the hierarchy: take two copies of the same theory — say ${\mathrm{ZFC}}_{\text{obj}}$ + first-order logic as the object language, and ${\mathrm{ZFC}}_{\text{meta}}$ + FOL as the metalanguage — and use the second to analyse the first. The intuition is using German to analyse English: two languages of comparable expressive power, one talking about the other. Can it be done?

This is in fact the orthodox setup — "the metatheory is ZFC" almost always means exactly this, with ${\mathrm{ZFC}}_{\text{obj}}$'s formulas and proofs Gödel-coded as sets inside ${\mathrm{ZFC}}_{\text{meta}}$. But the analogy holds perfectly for syntax and breaks in one precise place for truth.

Where it works — syntax and proof theory. German fully describes the grammar of English; no extra power is needed, since describing a language's syntax is a combinatorial task. Likewise ${\mathrm{ZFC}}_{\text{meta}}$ captures all of ${\mathrm{ZFC}}_{\text{obj}}$'s syntax and provability — indeed a weak arithmetic (PRA) already formalises "$\phi$ is an axiom," "$\pi$ proves $\phi$," "${\mathrm{ZFC}}_{\text{obj}}⊢\phi$." ZFC proves a great many metatheorems about ZFC this way. Full marks for the analogy here.

Where it breaks — truth and consistency. The cracks appear when German asserts two specific things about English.

• A truth predicate. German can contain "true-in-English," precisely because German $\ne$ English — the Tarskian point. Mirroring this, ${\mathrm{ZFC}}_{\text{meta}}$ can define satisfaction $\mathrm{Sat}(M,┌\phi ┐)$ for any set-sized model $M$. But for the intended class universe $V$ ("truth in the real universe of sets"), Tarski's undefinability theorem forbids it: ZFC cannot define truth for its own full language. The disanalogy is exact — German out-resources English in precisely the dimension needed (it has the predicate English lacks), whereas two copies of ZFC have identical strength, so ${\mathrm{ZFC}}_{\text{meta}}$ holds no advantage over ${\mathrm{ZFC}}_{\text{obj}}$.
• Consistency / "has a model." To do genuine semantics — to assert "${\mathrm{ZFC}}_{\text{obj}}$ has a model" — the metatheory must prove $\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})$. By Gödel's second incompleteness theorem, ${\mathrm{ZFC}}_{\text{meta}}=\mathrm{ZFC}$ cannot: a theory cannot prove the consistency of one of equal-or-greater consistency strength, and two copies of ZFC are equiconsistent. $${\mathrm{ZFC}}_{\text{meta}}⊬\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})⟹{\mathrm{ZFC}}_{\text{meta}}⊬\text{“}{\mathrm{ZFC}}_{\text{obj}}\text{ has a model.”}$$ Here the prover (left of $⊬$) is always the metatheory, while the $\mathrm{Con}(\cdot )$ argument is an arithmetic statement about the object theory's coded proofs. Because the two copies are the same theory of the same strength, $\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})$ and $\mathrm{Con}({\mathrm{ZFC}}_{\text{meta}})$ are literally the same sentence — which is exactly why this reduces to the textbook $\mathrm{ZFC}⊬\mathrm{Con}(\mathrm{ZFC})$.

The fix — a strictly stronger metalanguage. To recover the full "German analysing English" power (a truth predicate for the object language and a soundness/consistency proof), the metalanguage must genuinely exceed the object language — reproducing the way German exceeds English in the one relevant respect:

• ZFC + an inaccessible cardinal $\kappa$: then $V_{\kappa }$ is a set modelling ${\mathrm{ZFC}}_{\text{obj}}$, the metatheory proves $\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})$, and truth-in-$V_{\kappa }$ is definable.
• ZFC + a truth class (e.g. compositional $\mathrm{CT}[\mathrm{ZFC}]$): adds exactly the missing "true-in-the-object-language" predicate.

The moral: two identical copies of ZFC+FOL work beautifully as object/metalanguage for syntax and proof theory, but "German" gains real semantic authority over "English" — a truth predicate, a consistency proof — only by being strictly richer. The analogy's hidden assumption is that the metalanguage out-resources the object language in the relevant dimension; equal-strength formal copies do not.

A reflexive caveat: asserting this needs a third level

The claim just made — ${\mathrm{ZFC}}_{\text{meta}}⊬\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})$ — is itself a statement about what ${\mathrm{ZFC}}_{\text{meta}}$ proves, and any assertion of the form "$T⊬\phi$" lives one level above $T$. So stating it as true puts us at a third level, a meta-meta-language treating ${\mathrm{ZFC}}_{\text{meta}}$ as its object of study. This remark thereby demonstrates its own thesis reflexively: provability- and model-talk always ascend a level, including talk about the metatheory. Three points keep this from being a vicious regress.

• Higher in role, not in strength. The third level is foundationally cheap. Gödel's second theorem in its honest conditional form, $\mathrm{Con}(\mathrm{ZFC})\to (\mathrm{ZFC}⊬\mathrm{Con}(\mathrm{ZFC}))$, is an arithmetic statement about provability predicates already provable in weak arithmetic (PRA) — far below ZFC. We climb a notch in role (talking about level 1) without climbing in consistency strength. Contrast the semantic tasks above (define truth-in-$V$, prove "has a model"), which genuinely need more strength: underivability is syntactic and cheap, existence-of-a-model is semantic and expensive.
• The unconditional assertion smuggles in a posit. The flat "${\mathrm{ZFC}}_{\text{meta}}⊬\mathrm{Con}({\mathrm{ZFC}}_{\text{obj}})$" is warranted only assuming $\mathrm{Con}(\mathrm{ZFC})$ — an inconsistent ZFC would prove everything, including its own $\mathrm{Con}$. That consistency assumption is a level-3 commitment the lower levels cannot supply for themselves (Gödel 2 again). What justifies that assumption in turn is taken up in What Justifies Con(ZFC)?.
• The buck stops at informal reasoning. This is the benign regress once more — potential, finite-on-demand, each step costing only weak arithmetic. At the top sits the informal mathematical English we actually reason in, never itself fully formalized: the working mathematician's ultimate metalanguage, where every formal "$T⊬\phi$" is finally underwritten.

The one move that is never available is collapsing all three levels into one so that ${\mathrm{ZFC}}_{\text{meta}}$ certifies its own non-self-provability of $\mathrm{Con}$ — exactly the collapse Tarski and Gödel forbid.

Semantics in mathematics

What does this mean for how meaning works in ordinary mathematics? Several observations.

• Mathematical semantics is set-theoretic semantics. To give the meaning of a formal language — what its terms denote and when its sentences are true — the standard tool is a structure: a set-with-interpretations, and Tarski's recursive satisfaction definition on top. "Semantics," in the mathematician's sense, just is this assignment of set-theoretic denotations. (Compare the alternative categorical / type-theoretic semantics, where meaning is given by functors into a category or by the term model of a type theory — but these too live in a metatheory.)
• The metatheory is where meaning is conferred. A formal system, considered purely as an object theory, is meaningless string-manipulation; its symbols acquire reference only when interpreted in a model, an act performed in the metalanguage. Syntax is intrinsic to the system; semantics is always conferred from outside. This is the precise content of "model talk is metalinguistic."
• Proof vs. truth, internal vs. external. Inside the object theory one has only $⊢$ (derivability); "is true" is an external, metalinguistic verdict relative to a chosen model. The completeness theorem is the bridge that makes these extensionally agree for first-order logic ($⊢\phi ⟺\models \phi$) — a reassurance that the internal, syntactic notion and the external, semantic one coincide. Incompleteness is the residual gap: relative to a fixed theory, provability falls short of truth-in-the-standard-model.
• "Standard model" is itself a metatheoretic commitment. Speaking of the natural numbers, or the cumulative hierarchy, presupposes a background universe in which that intended structure is singled out. By Skolem's paradox and the Löwenheim–Skolem phenomenon, no first-order object theory pins its intended model down; the "standardness" is supplied by the metatheory, not the theory.

Take-aways

• Tarski's hierarchy stratifies language into levels $L_0\subset L_1\subset \cdots$, each defining truth only for those below; the Liar dissolves because there is no level-free truth predicate.
• The resulting regress is benign: potential not actual, usually collapsed by taking ZFC as a universal metatheory, and optionally extended transfinitely or replaced by Kripkean fixed-point truth.
• A theory can largely be its own metatheory (two copies of ZFC+FOL, "German analysing English") for syntax and proof theory, but not for truth or consistency: by Tarski and Gödel 2, an equal-strength metatheory cannot define truth for the object language's whole universe or prove it has a model. Full semantic authority requires a strictly stronger metatheory (e.g. ZFC + an inaccessible).
• In mathematics, semantics = interpretation in a structure, an act performed in the metalanguage; syntax is internal, meaning is conferred from outside, and the completeness theorem certifies that the internal ($⊢$) and external ($\models$) notions agree for first-order logic.

Related discussions: Models, Truth, and Metalanguage (the single-step object/meta distinction and Tarski undefinability), Set Theory and First-Order Logic (foundational ordering), What the Metalanguage for First-Order Logic Assumes (the reverse-mathematics ledger, the axiom-of-choice subtlety, and why $\mathrm{Con}(\mathrm{ZFC})$ is not assumed), and Category Theory (an alternative account of mathematical semantics).