Non-standard model of arithmetic

Inmathematical logic,anon-standard model of arithmeticis a model offirst-order Peano arithmeticthat contains non-standard numbers. The termstandard model of arithmeticrefers to the standard natural numbers 0, 1, 2,…. The elements of any model of Peano arithmetic arelinearly orderedand possess aninitial segment isomorphicto 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 numbersthemselves.^[1]

Existence

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

From the compactness theorem

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 numeraln,the axiomx>nis included. Any finite subset of these axioms is satisfied by a model that is the standard model of arithmetic plus the constantxinterpreted 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 toxcannot 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 theoremfails. It can be proved inZermelo–Fraenkel set theorythat Goodstein's theorem holds in the standard model, so a model where Goodstein's theorem fails must be non-standard.

From the incompleteness theorems

Gödel's incompleteness theoremsalso 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 thatGis false in some model of Peano arithmetic. However,Gis true in the standard model of arithmetic, and therefore any model in whichGis false must be a non-standard model. Thus satisfying ~Gis a sufficient condition for a model to be nonstandard. It is not a necessary condition, however; for any Gödel sentenceGand any infinitecardinalitythere is a model of arithmetic withGtrue and of that cardinality.

Arithmetic unsoundness for models with ~Gtrue

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

From an ultraproduct

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, $\mathbb {N} ^{\mathbb {N} }$ .Choose anultrafilteron $\mathbb {N}$ ,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 resultingsemiringis a non-standard model of arithmetic. It can be identified with thehypernaturalnumbers.^[2]

Structure of countable non-standard models

Theultraproductmodels are uncountable. One way to see this is to construct an injection of the infinite product ofNinto the ultraproduct. However, by theLöwenheim–Skolem theoremthere must exist countable non-standard models of arithmetic. One way to define such a model is to useHenkin semantics.

Anycountablenon-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 bedenseand linearly ordered without endpoints, andthe order type of the rationals is the only countable dense linear order without endpoints.^[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) elementuis in the model, then so ism⋅ufor anymin the initial segmentN,yetu²is larger thanm⋅ufor any standard finitem.

Also one can define "square roots" such as the leastvsuch thatv²> 2 ⋅u.These cannot be within a standard finite number of any rational multiple ofu.By analogous methods tonon-standard analysisone can also use PA to define close approximations to irrational multiples of a non-standard numberusuch as the leastvwithv> $π$ ⋅u(these can be defined in PA using non-standard finiterational approximations of $π$ even though $π$ 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 theoremshows 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 iscomputableon the codes. This result was first obtained byStanley Tennenbaumin 1959.

References

Citations

^Hans Hermes (1973).Introduction to Mathematical Logic.Hochschultext. London: Springer.ISBN 3540058192.ISSN 1431-4657.Here: Ch. VI.3
^Goldblatt, Robert(1998), "Ultrapower Construction of the Hyperreals",Lectures on the Hyperreals,New York: Springer, pp. 23–33,doi:10.1007/978-1-4612-0615-6_3
^Andrey Bovykin and Richard KayeOrder-types of models of Peano arithmetic: a short surveyJune 14, 2001
^Andrey BovykinOn order-types of models of arithmeticthesis submitted to the University of Birmingham for the degree of Ph.D. in the Faculty of Science 13 April 2000
^Fred Landman LINEAR ORDERS, DISCRETE, DENSE, AND CONTINUOUS– includes proof thatQis the only countable dense linear order.

Sources

Boolos, George,andJeffrey, Richard1974.Computability and Logic,Cambridge University Press.ISBN 0-521-38923-2.
Skolem, Thoralf (1934)."Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen"(PDF).Fundamenta Mathematicae(in German).23(1): 150–161.doi:10.4064/fm-23-1-150-161.