Non-standard model of arithmetic

In mathematical logic, a nonstandard model of arithmetic is a model of (first-order) Peano arithmetic that contains nonstandard numbers. The standard model of arithmetic consists of the set of standard natural numbers {0, 1, 2, …}. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A nonstandard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

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 the compactness 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 symbol x. The axioms consist of the axioms of Peano arithmetic P together with another infinite set of axioms: for each numeral n, the axiom x > n is included. Any finite subset of these axioms is satisfied by a model which is the standard model of arithmetic plus the constant x 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 to x cannot be a standard number, because as indicated it is larger than any standard number.

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

From the incompleteness theorems

Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic. The incompleteness theorems show that a particular sentence G, the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic. By the completeness theorem, this means that G is false in some model of Peano arithmetic. However, G is true in the standard model of arithmetic, and therefore any model in which G is false must be a nonstandard model.

~G is a sufficient condition for a model, but not a necessary condition of course. For any such Gödel sentence G, there are models of arithmetic with G true of all cardinalities.

Arithmetic unsoundness for models with ~G true

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

From an ultraproduct

Another method for constructing a non-standard model of arithmetic is via an ultraproduct. A typical construction uses the set of all sequences of natural numbers, $\mathbb {N} ^{\mathbb {N} }$. Identify two sequences if they agree for a set of indices which is a member of a fixed non-principal ultrafilter. The resulting ring is a non-standard model of arithmetic. It can be identified with the hypernatural numbers.

Structure of countable non-standard models

The Ultrapower models are uncountable (since they are based on an infinite product of Z, so infinite sequences of numbers). By the [[

Löwenheim–Skolem theorem]] they must have a countable model - one way to do that is using Henkin semantics.

Any countable nonstandard model of arithmetic has order type ω + (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 the rational numbers. In other words, a countable nonstandard model begins with an infinite increasing sequence (the standard elements of the model). This is followed by a collection of "blocks," each of order type ω* + ω, the order type of the integers. These blocks are in turn densely ordered with the order type of the rationals.

Although the order type of the countable nonstandard models is known, the arithmetical operations are much more complicated.

Only the order type is the same. For instance given any nonstandard u then you can define the least v such that v²>u which isn't any rational multiple of u. Similarly you can define the decimal expansion of pi in PA, so can define the least v such that v> u * pi, and so on. Also if u is in the model, then so is m*u for any m, n in the initial segment N, yet u^2 is larger than m*u for any standard finite m. So the arithmetical structure of a countable nonstandard model is more complex than the rationals.

It is even more complicated than that though. Tennenbaum's theorem shows that there is no countable nonstandard model of Peano arithmetic in which either the addition or multiplication operation is computable. This result, first obtained by Stanley Tennenbaum in 1959, places a severe limitation on the ability to concretely describe the arithmetical operations of a countable nonstandard model.

References

• Boolos, G., and Jeffrey, R. 1974. Computability and Logic, Cambridge University Press. ISBN 0-521-38923-2
• Skolem, Th. (1934) Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23, 150–161.