Axioms of arithmetic and isomorphism

Question

What does it mean to say that the axioms of arithmetic (Peano's or Dedekind's) are sufficient to characterize the abstract, mathematical structure of the natural numbers up to isomorphism?

Answer 1

When we say that the axioms are sufficient to describe a structure up to isomorphism we mean that any two structures in which the axioms hold are isomorphic.

To be more explicit, let $\cal L$ be a language, and $T$ a theory in some¹ logic. Then $T$ has a unique model up to isomorphism if whenever $M$ and $N$ are two $\cal L$ structures such that $M\models T$ and $N\models T$, then $M\cong N$.

For example, if we consider the axioms of an ordered field (the usual axioms of a field, plus the axioms of a linear order, plus the axioms stating that the operations behave nicely with the order) and the second-order logic axiom stating "Every bounded set has a least upper bound", then every model of the theory is isomorphic to $\mathbb R$ (and the isomorphism is unique in this case).

Another example is the Peano axioms with second-order induction, every model of this theory has to be isomorphic to the natural numbers.

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Footnotes:

1. I wrote in some logic, because in first-order logic if there is an infinite model for $T$ then there is a model of any infinite cardinality. This means that there are many non-isomorphic models.

   We can, however, salvage some characterization, and some theories are nice enough that they have the property that for some $\kappa$, all the models of cardinality $\kappa$ are isomorphic.

   For example the theory of a dense linear order without endpoints have this property for $\kappa=\aleph_0$. In particular every linear order which is countable and dense and does not have maximum and minimum is isomorphic to the rational numbers.

   Similarly the theory of algebraically closed fields of characteristics zero has this property for every $\kappa>\aleph_0$. So any algebraically closed field which contains the rational numbers and have cardinality $2^{\aleph_0}$ is isomorphic to the field of the complex numbers.