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

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.

Using Gödel's original "Gödel numbering" system, one can, for any first-order statement, come up with a corresponding new statement asserting various numeric relationships between ordinary natural numbers - effectively creating an imitation of PA within PA itself. One can use this system to encode very complex metamathematical propositions, such as the notion of a sequence of statements being a "valid proof" of a theorem from the PA axioms, or the notion of one sequence of statements being a proof of another, final statement, or of a valid proof existing of a certain statement at all. These statements can also be encoded as statements asserting the existence of certain equality and prime factorization relationships on the natural numbers. The "G" sentence is formed from the initial proposition that "no sequence of statements exists which is a proof of G." When translated using Gödel's numbering system, this effectively becomes a statement postulating that no natural number exists which satisfies a certain set of these numeric relationships.

It can correspondingly be shown that no standard number satisfies the numeric conditions that correspond to the G-proof-encoding numeric property in question. However, one can construct a model in which the Gödel sentence is false, thus asserting there is indeed a necessarily nonstandard "number" which is defined as simply satisfying the abstract numeric relationships in question, despite that this number loses the interpretation of encoding an actual proof of something. In these models, although such a number is defined as existing, it simply satisfies these numeric relationships without in any way correlating to an actual proof of G.

The same applies to Gödel's second incompleteness theorem, which asserts that PA is undecided on whether there may exist numbers which "encode a proof" of statements which would conflict with the PA axioms, such as 1=0. There are necessarily nonstandard models of PA in which such numbers do exist - however, as with G, these numbers are simply defined as satisfying the set of numeric relationships corresponding to the Gödel encoding of the metamathematical propositions in question, without having any bearing on whether the actual metamathematical propositions are true.

It is noteworthy that the existence of the above models doesn't imply that they're inconsistent: while statements such as 0=1 which are provably false in PA are false in every nonstandard model, the above models show that particular systems of Gödel numbering may map these false statements onto propositions which nonstandard numbers satisfy, rather than which no numbers of any kind satisfy. The above formal systems are sometimes called ω-inconsistent, which should be noted is not the same as formal inconsistency: a theory being ω-inconsistent simply means that it's not satisfied by the standard model of arithmetic (and hence that only non-standard models of it exist).

It's also notable that while PA + ~G is a concrete example of a theory with only nonstandard models of it, that this not the most common or useful way to generate non-standard models of PA. Other axioms are instead often used to assert the existence of a non-standard number in the theory and so force its models to be non standard. Note also that there exist non-standard models for PA + G, of all cardinalities, as well, but the difference is that PA + ~G has only nonstandard models whereas PA + G is supported by the standard model as well.

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

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. 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.