Models of Peano Arithmetic and Non-Standard Models

Core Idea

Peano arithmetic (PA) has non-standard models: countably infinite models satisfying all PA axioms but containing infinite integers beyond all standard numerals. Every non-standard model contains a copy of the standard natural numbers followed by a densely ordered structure of infinitely large elements. Non-standard models demonstrate that first-order logic cannot axiomatize arithmetic uniquely.

How It's Best Learned

Construct a non-standard model using the compactness theorem by adding a constant c and axioms c > n for all numerals n. Study the structure of the infinite part.

Explainer

From your study of formal arithmetic and first-order logic, you know that Peano Arithmetic (PA) is a first-order theory with axioms for zero, successors, addition, and multiplication, plus an induction schema. You may have hoped these axioms uniquely pin down the natural numbers ℕ. The existence of non-standard models is the fundamental theorem showing they do not — and cannot.

The construction of a non-standard model is a direct application of the compactness theorem. Extend the language of PA with a new constant symbol c, and add the sentences c > 0, c > 1, c > 2, ... for every standard numeral. Each finite subset of these axioms is satisfiable (interpret c as a sufficiently large standard number). By compactness, the entire extended theory is satisfiable, producing a model M in which c is an "infinite integer" — greater than every standard natural number, yet satisfying all PA axioms. The elements corresponding to standard natural numbers form an initial segment isomorphic to ℕ, but M contains additional non-standard elements beyond this segment.

The structure of the non-standard part is illuminating. Every non-standard element z satisfies z > n for all standard n, yet z − 1, z − 2, ... are also elements of M, stretching infinitely in both directions within the non-standard region. The non-standard elements form a densely ordered collection of copies of ℤ — each "block" is isomorphic to the integers, and the blocks themselves have no least or greatest element. This contrasts sharply with the discrete, well-ordered structure of the standard naturals. PA's induction schema does not rule this out, because first-order induction only quantifies over properties *expressible in first-order logic* — and first-order logic cannot single out the standard model from among all its non-standard cousins.

The philosophical consequence is profound: first-order logic cannot categorically axiomatize arithmetic. No matter what first-order sentences you add to PA (as long as they are all true in ℕ), the resulting theory will still have non-standard models. This follows from the Löwenheim-Skolem theorem and compactness: any first-order theory with an infinite model has models of every infinite cardinality, and even among countable models, non-standard ones exist. The "true arithmetic" — the set of all first-order sentences true in ℕ — is not recursively axiomatizable (by Gödel's incompleteness theorem), and non-standard models witness exactly this gap: they satisfy every provable sentence but disagree with ℕ on some unprovable truths.