True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?

Thanks!

Solution 1:

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.

The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.

Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.

It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.

In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.

One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.

To read more:

1. Completion of rational numbers via Cauchy sequences
2. question about construction of real numbers
3. Constructing $\mathbb R$
4. Why does the Dedekind Cut work well enough to define the Reals?
5. Construction of $\Bbb R$ from $\Bbb Q$
6. In set theory, how are real numbers represented as sets?