What the nonstandard models actually are
The Descent proved they exist: by compactness, first-order arithmetic has models full of infinite numbers, and ℕ cannot be singled out. But existence is not anatomy. The impostors are not a formless swarm — they have a forced shape, a coded interior, a recursion-theoretic floor, and a richness that lets them do what ℕ provably cannot. This is their dissection.
two origins · the forced shape · overspill · the coded interior · saturation · the self-modelling model
An impostor is a model of arithmetic that is not ℕ. The Descent showed that first-order arithmetic cannot keep them out: adjoin to the axioms a new constant c together with c > 0, c > 1, c > 2, …, and every finite subset is satisfiable in ℕ itself — so by compactness the whole is satisfiable, in a model with an element above every numeral. The intended structure cannot be fixed from inside. That was a proof of presence. It said nothing about form.
It would be natural to imagine the impostors as featureless — ℕ with some vague cloud of extra junk bolted on, too lawless to describe. The opposite is true. A nonstandard model of PA is one of the most rigidly organized objects in logic. Its order has a single forced shape. Its elements code an entire family of sets of standard numbers, and that family obeys precise closure laws. Its arithmetic cannot be carried out by any machine. And if it is countable, it is so saturated with realized possibilities that it can build, inside itself, a notion of truth for its own sentences — a feat ℕ's definable resources are forbidden by Tarski.
This panel dissects the impostor. We trace it from its two births, through its forced skeleton, into the coded organs of its interior, and out to the strange capacities of its richest specimens. The thesis of The Descent — that ℕ is unreachable yet, in deeper senses, anchored — is not a paradox once the anatomy is on the table. It is a description of an organism. We proceed in six movements.
There are two ways to manufacture a nonstandard model, and the difference between them is the difference between agreeing with ℕ and contradicting it.
Not every impostor is born the same way, and the distinction governs everything downstream. One lineage descends from compactness: take true arithmetic Th(ℕ) — the complete theory of the standard numbers, which already decides every sentence — and force a model to contain an infinite element. The result agrees with ℕ on every first-order sentence and is still not isomorphic to it. The other lineage descends from incompleteness: take PA + ¬Con(PA), consistent by Gödel's second theorem, and any model of it must disagree with ℕ — it satisfies a false arithmetic sentence, harbouring a coded "proof of 0=1" of nonstandard length. The first kind is an elementary extension of nothing it betrays; the second is a defector.
A model of true arithmetic with an infinite number. It cannot be told from ℕ by any first-order question; only its shape gives it away. No incompleteness is used — the complete theory has these models anyway.
A model that proves its own theory inconsistent — by exhibiting a nonstandard derivation of contradiction. It believes a falsehood about the numbers. The diagonal put it there; PA ⊬ Con(PA) is what makes it consistent.
Neither lineage is rare. Already Th(ℕ), complete as it is, has 2ℵ₀ pairwise non-isomorphic countable models; PA has continuum-many too. And the count is never small for a structural reason: by Vaught's theorem no first-order theory has exactly two countable models up to isomorphism — the number is 1, or ℵ₀, or 2ℵ₀, and for arithmetic it is the largest. The impostors are not exceptions slipped past the axioms. They are a continuum, and ℕ is one point in it, distinguished by nothing first-order logic can state.
The vocabulary matters for what follows. A nonstandard element is one larger than every numeral; the standard part is the initial segment of numerals, an isomorphic copy of ℕ sitting at the bottom of every impostor. The loyal impostors are exactly the elementary extensions of the standard part — they extend it without disturbing a single first-order truth; the defectors are extensions that break one. Both share the skeleton we turn to now, because the skeleton is forced by PA alone, and both lineages obey PA.
Every countable impostor has the same order type — ℕ followed by ℚ-many copies of ℤ. The shape is not chosen; it is forced by the axioms.
Begin with the bones. Ask not what an impostor contains but how its elements are ordered, and the answer is astonishingly rigid: up to order-isomorphism there is exactly one possibility for a countable nonstandard model. The standard part ℕ sits at the bottom as an initial ray of type ω. Above it, the nonstandard elements arrange themselves into two-way-infinite chains — copies of ℤ — and those chains are packed together as densely as the rationals, with no first chain and no last. The whole order is ℕ + ℤ·ℚ. Zoom in by stages and watch each feature become unavoidable.
The shape at full scale: a discrete ray, then — past the horizon no numeral reaches — a continuum-dense body. There is no largest standard number and no smallest nonstandard one; the two regions meet across a gap, not a seam.
PA proves every nonzero element has a unique predecessor, so a−1, a−2, … all exist; and each stays nonstandard, since if a−k were standard then a = (a−k)+k would be too. Going up and down by one, a generates a full copy of ℤ that never touches the standard ray.
Given chains below a < b, the difference b−a is nonstandard, and PA's division by two yields a midpoint m a nonstandard distance from both — hence in a new chain between them. Between any two chains lies a third: the chains are densely ordered, with no endpoints, which for a countable order means a copy of ℚ.
the skeleton is only the skeleton Order, though, is the cheap part. The shape ℕ + ℤ·ℚ is shared by continuum-many models that are not isomorphic as models of arithmetic — they agree on how their elements line up while disagreeing on the sums and products laid over that line. Two impostors can carry the identical order and still satisfy different theories, because addition and multiplication add structure the ordering does not see. The order type is forced and uniform; the arithmetic on top is where the impostors become individuals. To see how much they hide, we stop looking at the line and start asking what their elements encode.
Inside an impostor, no formula picks out exactly the standard numbers. Any property that catches them all overshoots — into the nonstandard.
The standard part ℕ is plainly there inside every impostor, sitting at the bottom of the order. The question is whether M can see it — whether any formula φ(x), with parameters from M, holds of exactly the standard elements. It cannot, and the reason is a one-line consequence of induction. Suppose φ defined precisely ℕ. Then its negation would define the nonstandard elements M ∖ ℕ — a nonempty set. PA proves the least-number principle for every definable set, so M ∖ ℕ would have a least member. But it has none: if a is nonstandard, so is a−1, which is smaller. Contradiction. The standard cut is invisible to the model's own language.
a familiar shape This is The Descent's Tarski, refracted through a single model. There, no first-order formula defined arithmetic truth, on pain of the Liar; here, no first-order formula defines the standard cut, on pain of a least infinite number. Both are the same impossibility — a structure cannot, from inside its own definable resources, draw the line between itself and the standard core it contains. The impostor cannot point to its own standard part any more than ℕ can point to its own truth. Undefinability is not a property of ℕ alone; it is a property of the relation between a model and the standardness it cannot articulate.
Far from a mere obstruction, overspill is the workhorse of the subject. Want to show some standard fact holds? Show its definable version holds of all standard numbers; overspill hands you a nonstandard witness, and you reason where infinite resources are available, then pull the conclusion back down. It is the engine behind constructions of satisfaction classes, behind conservativity proofs, behind the coding we turn to next: because the standard cut is invisible, anything M can define that looks bounded by the standard numbers must already extend beyond them — and that overhang is where the impostor stores its secrets.
A single element of an impostor codes a whole set of standard numbers. The family of all such sets obeys exact closure laws — and one of them must be uncomputable.
Here the impostor stops being a line and becomes a library. In PA one can code finite sets and sequences by single numbers — the n-th bit of a, written (a)n, is a definable function. Run that decoding on a nonstandard element and read off only the standard positions, and a names a set of ordinary natural numbers: { n ∈ ℕ : M ⊨ (a)n = 1 }. A nonstandard a has nonstandardly many bits, so it can code an infinite subset of ℕ — one of genuinely unbounded complexity. The collection of all sets coded this way is the standard system of the model, written SSy(M). It is the impostor's interior catalogue: every set of standard numbers it is able to hold in a single element.
The constraint is tight but not total: by a theorem of Scott (and Knight–Nadel), every countable Scott set is the standard system of some nonstandard model of PA. The interior can be any self-consistent little set-universe — but always one containing the uncomputable. And the loyal impostors mark the ceiling of arithmetic's own complexity: the theory they share, true arithmetic Th(ℕ), sits at Turing degree 0(ω) — above every finite level of the arithmetical hierarchy, the join of them all. To decide what is true in ℕ is to climb past every arithmetically definable question at once.
the floor, recovered This is where The Descent's second sense of "floor" comes from. The impostors deny ℕ a definitional floor — no first-order theory pins it. But Tennenbaum hands back a recursion-theoretic one: among all the models of PA, exactly one is computable, and it is ℕ. The very coding that fills the impostor's interior with uncomputable sets is what forbids the impostor from being computable itself. Unreachable by theory, ℕ is nonetheless singled out by computation — and the proof is not soft. It runs straight through the catalogue we have just read out of a single nonstandard bit-string.
The richest impostors realize every consistent possibility — and can carry, inside themselves, a notion of truth for their own sentences, including ones too long to be standard.
The deepest property an impostor can have is not size but saturation: the habit of realizing whatever it consistently could. A type over M is a set of conditions p(x) — formulas with parameters from M — that is finitely satisfiable, each finite part met by some element. M realizes p if a single element meets all of it at once. A model is recursively saturated when it realizes every recursive finitely-satisfiable type. Such models are not exotic — every consistent completion of PA has countable recursively saturated models, and every countable model has a recursively saturated elementary extension. These are the impostors filled to capacity: every possibility their theory permits is already present in some element.
how the bound is slipped Hold this against The Descent's Tarski. There, arithmetic truth was undefinable: no formula of ℕ captures the true sentences, on pain of the Liar. The impostor does not refute that — it circumvents it. It does not define its truth; it is saturated enough to carry truth as an external attachment, living among its nonstandard objects, outside the reach of its own definitions. What ℕ cannot do with its definable resources, a rich enough impostor can do by accepting an undefinable gift. The undefinability theorem closes one door — definition — and the saturated model walks through another.
The gift is not canonical, and the discipline that would make it so is exactly what is missing. Because the satisfaction class obeys no induction, it can be pathological: one can build a full satisfaction class that declares true a disjunction (0≠0 ∨ 0≠0 ∨ ⋯ ∨ 0≠0) of nonstandard length a — every disjunct false, the disjunction "true" — violating the plainest principle that a disjunction holds only if some disjunct does. The impostor's internal truth is compositional but need not be correct; truth without induction over truth is truth that can drift. So the impostor "knows itself" only in a loosened sense. It carries a Tarskian predicate ℕ is denied — and pays for the privilege in the coin of reliability. The bound was slipped, not broken.
An impostor begets impostors — both above itself, by conservative extension, and within itself, by building models of every theory it believes consistent.
The last organ is reproductive. An impostor is not a terminus; it generates further impostors in two directions at once. Upward, it can always be extended: a fresh layer of infinite numbers can be stacked above every element it has, with no first-order truth disturbed. Inward, it builds models itself: because PA proves the completeness theorem in arithmetized form, any impostor that believes a theory consistent contains a definable copy of a model of that theory. The impostor is at once a rung in an endless tower of models and a workshop assembling models on its own interior. There is no ground floor to the tower and no top — the same double absence The Descent found, here made concrete in the breeding of structures.
no ground floor, no top floor Put the two together and the breeding never settles. By MacDowell–Specker the model is never the tallest — there is always a conservative storey above. By arithmetized completeness the model is never the simplest — it codes models of theories within. The tower of impostors has no maximal rung and no minimal one; it is dense with structures that contain and are contained, each believing the next impossible or building it definably. ℕ is the unique well-founded point of this tower — the one model with no infinite descending chain of "numbers" — and exactly that well-foundedness is what no first-order axiom can demand. The impostors are the price of speaking about ℕ in a logic that cannot say well-founded.
The impostor is not chaos. It is a rigidly organized organism — and its organization is exactly why ℕ is unreachable yet anchored.
Step back from the table. The impostor that looked, at the start, like ℕ with formless junk attached has turned out to be one of the most lawful objects in mathematics: a single forced order, an interior that must obey the three Scott-set closures and must contain something uncomputable, a richness that can host truth, and a well-founded base it provably cannot name. Every feature is determined, and several are determined to be strange.
And this is precisely why The Descent's two verdicts are not in tension. ℕ is unreachable — the impostors are a continuum, first-order indistinguishable in the loyal case, and no theory or definable cut can fence them out; that is the missing definitional floor. Yet ℕ is anchored — it is the unique computable model, by the same coding that makes the impostors uncomputable, and the unique well-founded one, sitting inside every impostor as a base none of them can isolate; those are the recursion-theoretic and order-theoretic floors that remain. The anatomy of the impostor is the explanation of the paradox. The thing cannot be cornered by description because its rivals are everywhere — and it cannot be mistaken for any of them, because every rival carries, in its forced and coded body, the proof that it is not ℕ.
The impostor was meant to be the flaw — the proof that ℕ cannot be pinned. It is the flaw. But laid open, it is also the proof that ℕ survives the failure to pin it: organized to the last bit, every counterfeit carries the watermark that betrays it, and the genuine article is the one structure no description reaches and no rival resembles. No theory catches ℕ. Nothing else is ℕ. The impostors are how both stay true.