The rigorous treatment of analysis that we have seen requires a rigorous axiomatic treatment of numbers, in particular real numbers and a justification of the completeness axioms. We have seen axioms for the real numbers, in the form of the axioms of complete Archimedean ordered fields, but although these axioms are satisfactory and express our intuitions about the reals perfectly well, it would be nice to see mathematical arguments showing that a system of real numbers does indeed exist.
Obviously, such an argument should rely on some other axiomatic system, such as an axiomatic view of set theory, and we are potentially left with the issue of knowing why that set of axioms is a reasonable position to take. It is indeed possible to give an axiomatic presentation of set theory and within it a construction of the real numbers, though I won't go quite so far as that in these web pages. Instead I will give a rather more informal construction of the real numbers, one that will suffice to develop our intuition of what a real number is, and to provide strong evidence (though not necessarily a watertight proof) that such a system of numbers really does exist.
Before we can give a constructions of the real numbers, however, we must review the main building blocks: the natural numbers, integers and rationals. The next web page therefore starts a couple of steps further back and discusses the natural numbers and integers.
Before I start, I should present a general warning that the material here is quite long, and although most of it is not very difficult, and quite a lot is actually rather easy, it does have some tricky moments. It is somewhat more abstract that the other material on these web pages, and for this reason alone will probably be found quite hard-going by some readers. In addition, the section on the natural numbers might also require a bit of soul-searching (or navel-gazing) before you are convinced that it avoids a vicious circularity.
For all these reasons, I have put all these constructions of the main number systems at the end of this introductory web-course on pure mathematics, though properly it should come at the beginning. Most of this material is for interested readers only, and is not normally part of a first course in real analysis.
Furthermore, as previously mentioned it is impossible to provide fully rigorous axiomatic proofs of all these constructions in the context of this course, because the set-theoretic background required to do this is not available to most readers at this stage of their mathematical career. The arguments presented here are therefore given in a slightly less formal style than they perhaps deserve, and there are a few places, particularly in the material concerning the natural numbers, that should be explained further but cannot without assuming additional background material.
So rather than giving a complete and perfect account (if such a thing could ever exist) I have aimed to give a presentation that can be understood at this stage and which can be satisfactorally formalised without any problems at a later stage. As well as this, I hope I also give a good flavour of the arguments.
Finally, at the time of writing, time constraints have meant that at present
I have been unable to complete all the proofs. So there are a lot of
rather dull proofs to be filled in, which are currently simply marked
as exercises
. Some of these will remain as exercises in the
future (they are not hard). The remainder will be typed up when I have
time. I'm sorry that that time isn't right now.