True Arithmetic (Th(ℕ))

Origin. The object is implicit in Gödel (1931) — the set of true sentences is what PA fails to exhaust — and made explicit by Tarski's undefinability theorem (1933, published 1936), which shows the set is not arithmetically definable. Its recursion-theoretic location, degree 0^(ω), follows from the Kleene arithmetical hierarchy and Post's theorem.

Models. Everything true of the standard model, taken as a theory. Th(ℕ) is complete by construction, consistent, and not effectively axiomatizable — the last is Gödel's theorem restated. It is the theory PA is trying to be, and the gap between them is the subject of most of the metatheory in this collection. As a theory it is well-defined and useless; as a benchmark it is where every incompleteness result is measured from.

Formalism.

Language: 0, S, +, ×, < — PA's.

Definition: Th(ℕ) = {φ : ℕ ⊨ φ}, the set of first-order sentences true in the standard model.

Axiomatization: itself. There is no other, and there is no effective one.

Properties, each an immediate consequence of the definition or a named theorem: Complete: for every sentence φ, either φ ∈ Th(ℕ) or ¬φ ∈ Th(ℕ). (Definition.) Consistent: it has a model. (Definition.) Not recursively axiomatizable: by Gödel's first theorem, since a consistent effectively axiomatized extension of Q is incomplete. Not arithmetically definable: by Tarski's undefinability theorem — there is no formula True(x) with ℕ ⊨ φ ↔ True(⌜φ⌝) for all φ. Turing degree: 0^(ω), the degree of ⊕ₙ ∅^(n), by Post's theorem — the Σₙ fragment Th_Σₙ(ℕ) has degree ∅^(n), and Th(ℕ) is their effective join.

Fragments, by quantifier complexity: Σ₁-Th(ℕ): r.e., degree 0′; provable in Q (Σ₁-completeness). Σₙ-Th(ℕ): degree 0^(n). Th(ℕ): degree 0^(ω) — above every 0^(n) and below 0^(ω+1).

Non-standard models: every consistent completion of PA other than Th(ℕ) has only non-standard models; Th(ℕ) itself has non-standard models too, by compactness — completeness does not give categoricity. All countable models of Th(ℕ) are elementarily equivalent to ℕ and almost none are isomorphic to it (there are 2^ℵ₀ of them up to isomorphism).

Relation to the collection's other objects: PA ⊊ Th(ℕ), the inclusion strict by Gödel. Th(ℕ) is a completion of PA; PA has 2^ℵ₀ completions, of which Th(ℕ) is one, and the only one that is true. Ω-logic, second-order arithmetic, and the arithmetical hierarchy each supply a different apparatus for approaching it from below.

Symbols.

Metatheory. Th(ℕ) is the limit object of the arithmetical hierarchy, and its two defining facts are negative: it is not effectively axiomatizable (Gödel) and not arithmetically definable (Tarski). These are different theorems and it is worth keeping them apart — the first says no machine enumerates it, the second says no formula in its own language picks it out, and the second is the stronger. Together they mean the theory exists as a set-theoretic object and as nothing more usable than that. Every completion of PA is a candidate for Th(ℕ) and PA proves nothing that distinguishes the true one; the whole apparatus of axiomatic truth theories, reflection principles, and ordinal analysis is the attempt to climb toward it by effective means, and 0^(ω) is how far there is to climb.

Applies to. The statement of incompleteness — "true but unprovable" is a claim about membership in Th(ℕ) and non-membership in the theorems of PA. Degree theory, as the standard example at 0^(ω). Axiomatic truth theories, which approximate the undefinable predicate by axioms. Reflection principles and ordinal analysis, which measure how far a theory reaches into it. Model theory of arithmetic, where Th(ℕ) is the theory whose non-standard models are studied.

Limitations. Placement here is by the criteria and is defensible but marginal: Th(ℕ) has a signature and axioms in the letter of the definition, but the axioms are not given by a rule, and every other entry in this division is effectively presented. Nothing can be proved in it, only about it. It is not a foundation, not a formal system in the Hilbertian sense, and its use is entirely as a reference point. The phrase "true arithmetic" also invites a realism about ℕ that the formal object does not require and does not settle.

© 2026 Lingenic LLC