Axioms for sets of numbers

What is the most common axiomatic system used by modern mathematicans for the properties of the integers, rationals, reals, and complex numbers?

Or does one commonly use a single axiomatic system that is agreed apon, to construct axiomatic systems for the other sets of numbers. Such as using axioms of the integers to construct axioms for the rationals, or are there seperate cases for each individual set.

Also how does one know weather or not the axioms in each individual set will be consistant with one another, ie how does one know it is not possible to derive a contradiction from a given set of axioms from some axiomatic system representing a set of numbers.

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I think Peano arithmetic would be what you're looking for. But from what I know, it only defines the integers. – Joe Z. Dec 24 '12 at 2:07

Most practicing mathematicians, when dealing with all the number systems you mention, probably don't really care about the foundations and consistency issue.

To answer your questions: One can start with a model for set theory and then build inside that model a model for what is called Peano Arithmetic (PA). Basically, PA is a set of axioms about the basics of the system of natural numbers. Taken together with the principle of induction (which is not a first order notion) it can easily be shown that PA + Induction is categorical: any two models satisfying these axioms are isomorphic (meaning they are essentially the same).

With a model for PA+Induction one can construct, still within the model of set theory, a model of the integers, and then the rational, by considering an appropriate equivalence relation on sets of pairs of naturals (resp. integers). With a model for the rational now at hand, one can proceed (in several different ways) to complete it and obtain a model of the real numbers. Finally, considering pairs of real numbers, one can define a model for the complex numbers.

You will notice that all of these constructions have the common theme: From an existing model of some system X, construct (somehow) a model for system Y. This shows what is known as relative consistency: If system X is consisten (i.e., has a model) then system Y is consistent as well.

This works quite well but of course relies on the existence of a model for set theory. This is pretty much the end of the road though as it is a fundamental result in logic that a system rich enough to discuss set theory, if consistent can't prove its own consistency. So basically, we all believe that there exists a model for set theory.

There are some variations on the theme. For instance, instead of set theory as foundations one can use Topos theory. In a rich enough topos one can give analogous definitions of the natural, integer, rational, real, and complex numbers and, depending on the topos, these systems can be very different than the classical ones. For instance, there exists a topos in which every real valued functions $f:\mathbb R \to \mathbb R$ is continuous.

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I think you may be a bit confused. What do you mean by "axiomatic system used by modern mathematicians for the properties of the integers"? Do you mean the properties they have such as $a + (-a) = 0$ for all $a \in \mathbb{Z}$?

We are given ZFC. We then construct the integers. You can find this construction in books on set theory. Then, we consider rational numbers to be ordered pairs of integers (where the second entry is non-zero) under the obvious equivalence relation. To construct the real numbers, we look at all rational sequences and look at the equivalence classes of Cauchy sequences (two sequences $(a_n)$ and $(b_n)$ are equivalent if $a_n - b_n \to 0$). Complex numbers are pairs of real numbers under the obvious equivalence relation, and multiplication and addition are defined in the obvious way. We then verify that the rational, real and complex numbers are a field. We also then verify that the integers are an integral domain.

The algebraic properties of the real numbers, for instance, are just the axioms of a field. But each of these properties is just a theorem. Hence, if you believe ZFC is consistent, the axioms of a field will not lead to any contradictions.

Note that these axioms are far from sufficient to prove many important things. For instance, $\mathbb{C}$ is algebraically closed, but this cannot be proven from the axioms of a field. Indeed, we need complex analysis for that.

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it's worth noting that the completion of the rational numbers to obtain the reals can be achieved in at least two others ways: Dedekind cuts and ultraproducts. – Ittay Weiss Dec 24 '12 at 2:47