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 existence of such models is due to Thoralf Skolem (1934).

Existence

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 standard natural number 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 some largest number (named by 'x'), and 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, indeed as indicated it is a "largest" 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; because it can be proved in ZFC that Goodstein's theorem holds in the standard model, a model where Goodstein's theorem fails must be nonstandard.

Countable models

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 a countable nonstandard model is known, the arithmetical operations are much more complicated. 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.