Models, Truth, and Metalanguage

A model is a semantic object, and semantic talk — "satisfies," "true-in," "is a model of" — has a peculiar logical status: it can never live in the same language as the sentences it interprets. This remark makes that precise via Tarski's undefinability of truth and explains the object-level/meta-level "interplay" that results.

It relies on the object theory vs. metatheory distinction developed in Set Theory and First-Order Logic; read that first if the terms are unfamiliar. For the underlying definition of model and structure, see What is a model?.

Recap: object theory vs. metatheory

Briefly: when a formal system is the thing under study — a set of strings closed under derivation rules — it is the object theory. The reasoning we use to study it (defining its models, proving soundness/completeness) is the metatheory. The two may use the "same" notions (sets, formulas) but at different levels, and keeping them apart dissolves a great deal of apparent paradox. The remarks below are an application of that distinction to the specific notions of model and truth.

Model talk is always metalinguistic (Tarski undefinability)

A model is a semantic object — a structure $\mathcal{M}$ together with the satisfaction relation $\mathcal{M}\models \phi$. None of that vocabulary — "structure," "domain," "interpretation," "satisfies," "true-in" — belongs to the object language. The object language of ZFC contains only $\in$, variables, connectives, and quantifiers: it can write the sentence $\forall x\exists y(x\in y)$, but it cannot, in that same language, say "this sentence is true in $\mathcal{M}$." The predicate "is a model of" is irreducibly metalinguistic — it talks about object-language sentences and the structures satisfying them. Hence:

• manipulating formulas (proof, derivation, $⊢$) is object-level syntax;
• talking about models ($\models$, truth, satisfaction) is always meta-level semantics.

This is not mere convention. Tarski's undefinability of truth makes it a theorem: a sufficiently strong consistent theory cannot define its own truth predicate in its own language — otherwise a Liar sentence would yield a contradiction. So "true-in-the-intended-model," and with it "model," must ascend to a metalanguage strictly richer than the object language. Talking about models object-internally is not just avoided; in general it is provably impossible.

Two caveats keep this from being absolute:

• The metalanguage is itself formalizable, and then becomes an object theory. "Metalanguage" is a role, not a fixed thing. Proving completeness in ZFC makes ZFC the metatheory and first-order logic the object of study; studying that proof steps up again. The hierarchy is relative and re-entrant, not two fixed floors.
• Internalized model theory does not escape the metalevel. Set theory can treat models as object-level sets, with a satisfaction relation $\mathrm{Sat}(\mathcal{M},┌\phi ┐)$ defined on Gödel codes of formulas — this is how forcing and inner-model theory proceed. But the formulas had to be coded into sets first, and ZFC still cannot define truth for its own full language (Tarski again). The object/meta distinction survives internalization; it is merely relativized.

The interplay — one structure, two gazes

Object theory and metatheory are best read not as two different things but as the same objects switching roles with the vantage point. The very same first-order sentences are, from below, formal strings pushed around by proof rules; from above, claims interpreted in structures and evaluated as true or false. A model only ever appears in the second, looking-down posture:

So the object/meta relation is an interplay of how the model is looked at — a shift of gaze, not a change in the objects. This is exactly why the set-theory/first-order-logic "circularity" is a level confusion rather than a vicious circle: ZFC-as-strings and ZFC-as-where-models-live are the same symbols seen from two directions. Skolem's paradox is the vivid case — "uncountable" asserted inside the model, the bijection refuting it living outside in the metatheory: same structure, two gazes, no contradiction.

Take-aways

• Model talk is always metalinguistic. "Satisfies / true-in / is a model of" lives in the metalanguage, never the object language — and by Tarski's undefinability of truth it cannot be internalized in general.
• The object/meta relation is one structure seen from two gazes: strings from below, interpreted claims from above; a model appears only in the looking-down posture.
• The distinction is relative and re-entrant — any metalanguage can itself be formalized and studied as an object theory — and survives "internalized" model theory, which merely relativizes it via Gödel coding.

Related discussions: The Metalanguage Hierarchy and Semantics in Mathematics (climbing the object/meta tower upward, and what semantics means in practice), Set Theory and First-Order Logic (the object/meta distinction and foundational ordering), Skolem's paradox (absoluteness across models), and Gödel's incompleteness (a theory reasoning about a coded copy of its own syntax).