The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers.
Non-standard models of arithmetic exist only for the first-order formulation of the Peano axioms; for the original second-order formulation, there is, up to isomorphism, only one model: the natural numbers themselves.
The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem.
Any finite subset of these axioms is satisfied by a model that 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*.
Using more complex methods, it is possible to build non-standard models that possess more complicated properties.
Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic.
By the completeness theorem, this means that G is false in some model of Peano arithmetic.
, 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).
For instance if a nonstandard (non-finite) element u is in the model, then so is m ⋅ u for any m in the initial segment N, yet u2 is larger than m ⋅ u for any standard finite m. Also one can define "square roots" such as the least v such that v2 > 2 ⋅ u.
Once more, v − (m/n) ⋅ (u/n) has to be larger than any standard finite number for any standard finite m, n. [citation needed] This shows that the arithmetical structure of a countable non-standard model is more complex than the structure of the rationals.