Questions: Models of Peano Arithmetic and Non-Standard Models

Question 1

A student argues: 'If we take all first-order sentences that are true in the standard natural numbers ℕ and add them as axioms, the resulting theory will be categorical — its only model will be ℕ.' What does model theory say about this claim?

• A. The claim is correct — a complete first-order theory with a unique infinite model is categorical
• B. The claim fails: even adding all true sentences of ℕ produces a complete theory, but the Löwenheim-Skolem theorem guarantees it still has models of every infinite cardinality — including non-standard countable models
• C. The claim fails because no consistent first-order theory can have ℕ as a model
• D. The claim is correct for countable models, but uncountable non-standard models would still exist

Answer: (B)

The theory of 'true arithmetic' — all first-order sentences true in ℕ — is complete (every sentence is decided) but not categorical. The upward Löwenheim-Skolem theorem guarantees models of every uncountable cardinality, and compactness (plus downward L-S) guarantees countable non-standard models. Categoricity in first-order logic requires finiteness — no infinite structure is characterizable up to isomorphism by a first-order theory. The standard model ℕ is unique up to isomorphism only in second-order logic, not first-order logic. This is the fundamental limitation being exposed.

Question 2

Using the compactness theorem, a logician builds a non-standard model M of PA containing a non-standard element c greater than all standard naturals. What can be said about the element c + 1 in M?

• A. c + 1 = 0, since PA arithmetic wraps around at infinite elements — non-standard models have modular structure
• B. c + 1 is another non-standard element, greater than all standard naturals and greater than c
• C. c + 1 is undefined, since PA's successor function is only defined for finite elements
• D. c + 1 = c, since adding 1 to an infinite element leaves it unchanged

Answer: (B)

M is a model of PA, so all PA axioms hold in M — including the axiom that every element has a successor, and that the successor of x is x+1 > x. Since c > n for all standard naturals n, and c+1 > c, it follows that c+1 > n for all standard naturals as well. c+1 is a distinct non-standard element larger than c. The non-standard elements don't wrap around (that would violate the PA axiom that no element equals its own successor) and they don't collapse (that would violate the strict ordering axioms). The non-standard part forms a dense collection of ℤ-copies extending infinitely in both directions.

Question 3

Most model of Peano Arithmetic is isomorphic to the standard natural numbers ℕ.

• T. True
• F. False

Answer: False

This is the central misconception the topic addresses. Non-standard models of PA exist — they satisfy every PA axiom but contain elements greater than all standard naturals. The compactness construction (add constant c with axioms c > n for each standard numeral n) proves this directly. The standard naturals form an initial segment of any model of PA, but non-standard models extend beyond this with a dense collection of 'blocks' isomorphic to ℤ. First-order logic cannot categorically axiomatize ℕ — only second-order PA (with genuine set-quantification in induction) achieves categoricity.

Question 4

Non-standard models of Peano Arithmetic satisfy all theorems provable in PA, but they may disagree with ℕ on sentences that are true in ℕ but unprovable from PA.

• T. True
• F. False

Answer: True

A model of PA, by definition, satisfies every consequence of the PA axioms — including every provable theorem. Non-standard models are models, so they satisfy everything PA proves. The disagreement with ℕ occurs precisely on the sentences that PA cannot prove or disprove: unprovable truths of ℕ. Gödel's incompleteness theorem guarantees such sentences exist (in fact, there are true sentences of ℕ that no consistent recursively axiomatizable extension of PA can prove). Non-standard models witness this incompleteness — they are alternative models where those unprovable sentences happen to be false.

Question 5

Why can't the induction schema in Peano Arithmetic rule out non-standard models? What limitation of first-order logic does this expose?

Model answer: First-order induction only quantifies over properties expressible by first-order formulas in the language of PA. To rule out non-standard elements, you would need to express: 'the set of natural numbers is exactly the smallest set containing 0 and closed under successor.' But 'smallest set' requires second-order quantification over all subsets. First-order induction gives you: for each specific first-order formula φ, if φ(0) and φ(n) → φ(n+1), then ∀n φ(n). A non-standard element c satisfies all these individual induction instances, because no single first-order formula can define 'the standard naturals' in a way that excludes c.

This exposes the expressive limitation of first-order logic: it cannot quantify over all subsets of the domain, only over individual elements. The 'intended model' argument — that ℕ is the unique minimal model of PA — requires second-order reasoning about what the 'smallest' closure under successor is. First-order logic cannot capture minimality of infinite structures. This is why second-order PA (where the induction axiom genuinely quantifies over all subsets) is categorical for ℕ, while first-order PA is not. The gap between these two is the gap between categoricity and incompleteness.