Constructions of Familiar Number Systems

One of the things you learn like five times in the first half of your undergraduate math career is how to construct the reals from the rationals, the rationals from the integers, the integers from the naturals, the naturals from nothing. It’s an entertaining process and is a good way to fill up a lecture, and it might as well be reproduced here.

First, we begin life with , the empty set, and we define . (I do not typically endorse , but it fits in this construction so we’ll change conventions for this post.) To get at the rest of the natural numbers, we construct a successor function such that . Thus, , , , and so on. We induct, and we define the set given to us by induction to be , the set of natural numbers. We get a couple operators immediately; for instance, the minimum and maximum of a set of naturals is their intersection and union respectively. As shorthand for applying the successor function multiple times, we write , which gives us the usual definition of addition as an operation . Multiplication of , as you learned in grade school, is just adding to itself -fold times.

For those who have had any training in algebra before, smells like a group operation, but falls slightly short. It’s associative, the set is closed under it, we have an identity element, but we’re missing inverses. To this end, we consider the set of pairs of natural numbers, written . We define addition as component-wise and multiplication by the formula . Then, finally, we construct the equivalence relation if and only if , and define the quotient space to be . It’s up to the reader to check that we can carry over from and have them be well-defined. To recover the familiar notation for the integers, for any pair there must exist either a pair or in its equivalence class, and so to denote the equivalence class we write either or respectively.

Now that we have the structure of addition on the integers, we’ve constructed a ring, . We are missing the multiplicative inverses necessary to form a field, thus motivating the construction of the rationals. Let’s begin with an arbitrary commutative ring . We can induce a ring structure on by defining multiplication component-wise (i.e. ) and defining addition through the following formula: . Finally, we need another equivalence relation such that if and only if , and we construct the final ring as , for which we write as shorthand. is in fact a field, if has no zero divisors. Applying this construction to , we get , the set of rationals.

has most of the algebraic properties that we need for a basic structure to work in, but it’s missing an important analytic and topological one: completion. To this end, we need to switch perspective and think about metrics. We define a pre-metric on a set (which we will start calling a metric space, or just a space) to be a function such that the following properties hold:

• Nonnegativity:
• Indiscernibles:
• Symmetry:
• Triangle Inequality:

Some notes: A metric is a pre-metric whose codomain is . The reason we’re working with here is because we haven’t yet constructed the reals. Also, that last property is also sometimes called ’subadditivity’, a word that I personally have seen come up a lot more frequently in measure theory. Now, is naturally a metric space with metric , where denotes absolute value. We typically call this metric the Euclidean metric, since there are lots of viable metrics on that behave radically differently.

We haven’t talked about analysis yet because we haven’t had to use much of it, but I have been using the idea of sequences and convergence without ever defining them. To correct that, a sequence in is a function , and we typically write to mean the i^th term of the sequence, or to designate the sequence as a whole. We say that a sequence in a metric space converges to a point (called a limit point) if such that . The idea is that if we want to make the points in the sequence arbitrarily close to the limit point, then we just need to go past some finite point in the sequence, and then everything on that tail is within the desired distance. Now, we say a sequence is Cauchy if ) such that . The idea expressed in these quantifiers is that as we get farther out in the sequence, the individual terms of the sequence start to cluster up next to each other (but not necessarily around some particular point). All convergent sequences are Cauchy, but Cauchy sequences are not necessarily convergent in all metric spaces. For spaces where that reverse implication is true, we say that the space is ‘complete’.

There are a lot of definitions involving continuity and extension of uniformly continuous functions to complete spaces and topological density and thus at least point-set topology with its open balls and lots of other concepts. We can go back and revisit the particulars of the construction of at a later date, but suffice it to say that is the unique complete metric space such that every ball has nonempty intersection with . A useful corollary of the completeness of is that every set in with an upper bound has a least upper bound. For instance, the set did not have an upper bound in , but in that supremum exists and is called .

We have one last leap to make, one that I’ve already detailed before if you go back 30 or so posts. is not algebraically closed — that is, there exist polynomials with coefficients in with roots not in . In order to generate an algebraically closed field, we use the field extension , and we’re done.

Update: I was reading an introduction to K-theory last night and found that the construction of the integers from the naturals was reused in the context of k-rank vector bundles over a topological space.  Turns out math is interrelated.