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Inmathematical logic, anon-standard model of arithmetic is a model offirst-order Peano arithmetic that contains non-standard numbers. The termstandard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic arelinearly ordered and possess aninitial segmentisomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due toThoralf Skolem (1934).

Non-standard models of arithmetic exist only for the first-order formulation of thePeano axioms; for the original second-order formulation, there is, up to isomorphism, only one model: thenatural numbers themselves.^[1]

There are several methods that can be used to prove the existence of non-standard models of arithmetic.

The existence of non-standard models of arithmetic can be demonstrated by an application of thecompactness theorem. To do this, a set of axioms P* is defined in a language including the language of Peano arithmetic together with a new constant symbolx. The axioms consist of the axioms of Peano arithmetic P together with another infinite set of axioms: for each standard natural numbern, the axiomx >n is included. Any finite subset of these axioms is satisfied by a model that is the standard model of arithmetic plus the constantx interpreted as some number larger than any numeral mentioned in the finite subset of P*. Thus by the compactness theorem there is a model satisfying all the axioms P*. Since any model of P* is a model of P (since a model of a set of axioms is obviously also a model of any subset of that set of axioms), we have that our extended model is also a model of the Peano axioms. The element of this model corresponding tox cannot be a standard number, because as indicated it is larger than any standard number.

Using more complex methods, it is possible to build non-standard models that possess more complicated properties. For example, there are models of Peano arithmetic in whichGoodstein's theorem fails. It can be proved inZermelo–Fraenkel set theory that Goodstein's theorem holds in the standard model, so a model where Goodstein's theorem fails must be non-standard.

Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic.The incompleteness theorems show that a particular sentenceG, the Gödel sentence of Peano arithmetic, is neither provable nor disprovable in Peano arithmetic. By thecompleteness theorem, this means thatG is false in some model of Peano arithmetic. However,G is true in the standard model of arithmetic, and therefore any model in whichG is false must be a non-standard model. Thus satisfying ~G is a sufficient condition for a model to be nonstandard. It is not a necessary condition, however; for any Gödel sentenceG and any infinitecardinality there is a model of arithmetic withG true and of that cardinality.

Assuming that arithmetic is consistent, arithmetic with ~G is also consistent. However, since ~G states that arithmetic is inconsistent, the result will not beω-consistent (because ~G is false and this violates ω-consistency).

Another method for constructing a non-standard model of arithmetic is via anultraproduct. A typical construction uses the set of all sequences of natural numbers,${\displaystyle {\mathbb{N}}^{\mathbb{N}}}$. Choose anultrafilter on${\displaystyle \mathbb{N}}$, then identify two sequences whenever they have equal values on positions that form a member of the ultrafilter (this requires that they agree on infinitely many terms, but the condition is stronger than this as ultrafilters resemble axiom-of-choice-like maximal extensions of the Fréchet filter). The resultingsemiring is a non-standard model of arithmetic. It can be identified with thehypernatural numbers.^[2]

Theultraproduct models are uncountable. One way to see this is to construct an injection from the infinite product${\displaystyle {\mathbb{N}}^{\mathbb{N}}}$ into the ultraproduct. However, by theLöwenheim–Skolem theorem there must exist countable non-standard models of arithmetic. One way to define such a model is to useHenkin semantics.

Anycountable non-standard model of arithmetic hasorder typeω + (ω* + ω) ⋅ η, where ω is the order type of the standard natural numbers, ω* is the dual order (an infinite decreasing sequence) and η is the order type of therational numbers. In other words, a countable non-standard model begins with an infinite increasing sequence (the standard elements of the model). This is followed by a collection of "blocks", each of order typeω* + ω, the order type of the integers. These blocks are in turn densely ordered with the order type of the rationals. The result follows fairly easily because it is easy to see that the blocks of non-standard numbers have to bedense and linearly ordered without endpoints, and the order type of the rationals is the only countable dense linear order without endpoints (seeCantor's isomorphism theorem).^[3][4][5]

So, the order type of the countable non-standard models is known. However, the arithmetical operations are much more complicated.

It is easy to see that the arithmetical structure differs fromω + (ω* + ω) ⋅ η. For instance if a nonstandard (non-finite) elementu is in the model, then so ism ⋅u for anym in the initial segmentN, yetu² is larger thanm ⋅u for any standard finitem.

Also one can define "square roots" such as the leastv such thatv² > 2 ⋅u. These cannot be within a standard finite number of any rational multiple ofu. By analogous methods tonon-standard analysis one can also use PA to define close approximations to irrational multiples of a non-standard numberu such as the leastv withv >π ⋅u (these can be defined in PA using non-standard finiterational approximations ofπ even thoughπ itself cannot be). Once more,v − (m/n) ⋅ (u/n) has to be larger than any standard finite number for any standard finitem,n.^{[citation needed]}

This shows that the arithmetical structure of a countable non-standard model is more complex than the structure of the rationals. There is more to it than that though:Tennenbaum's theorem shows that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as (standard) natural numbers such that either the addition or multiplication operation of the model iscomputable on the codes. This result was first obtained byStanley Tennenbaum in 1959.

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