What are mathematical objects? The answer may surprise you. More on this story tonight.
Mathematical objects are mathematical objects.1 From a foundational point of view, we sometimes want to start with some atomic notion, and the argue that we can define the rest of the mathematical universe in terms of those objects.
These can be sets, as done in set theory, or it can be various types as done in type theory, and so on.
Ultimately, the goal is always to "reduce the existence to something more believable". Namely, if you believe that the rational numbers make sense, and that some basic constructions make sense (e.g. Cauchy sequences), then this is a proof why you should believe that the real numbers make sense.
Sure, you can now ask why do the rational numbers make sense. Then you can fallback to the integers, then to the natural numbers, and you can just accept that, or fall onto the empty set as done in the standard constructions in set theory.
But it's always something of the form:
- If you agree with me on the validity of this object, and
- you agree with me on the validity of that method, then
- you agree with me on the validity of this new object.
So the real numbers can be equivalence classes of Cauchy sequences, because that is one way of building the real numbers. Or the real numbers can be Dedekind-cuts, or non-empty proper-initial segments. Or any other thing.
The important thing, however, is that we can prove they are all "the same". Namely, if you construct the real numbers using one method, and I construct the real numbers using a different method, then there is a structure-preserving way to identify the two versions of the real numbers.
So, are real numbers equivalence classes of Cauchy sequences of rational numbers? are they Dedekind cuts of rational numbers? Are they sets, or types, or some category? Maybe they are atomic to mathematics just like the natural numbers, and so the real numbers are just that, "the real numbers"?
The answer is that it doesn't matter. As long as they satisfy the properties we "expect" the real numbers to satisfy.
Let me just finish by pointing out that Tao doesn't really suggest that the real numbers are limits of these Cauchy sequences. Limits are only defined within a particular space (e.g. $0$ is not the limit of $\frac1n$ in the space $(0,1)$, simply because $0$ is not a point in that space).
But Tao is preparing the ground for proving that every real number is the limit of a Cauchy sequence of rational numbers. But at that point, this is just a notation of the real numbers which is defined from a certain Cauchy sequence.
- I said it may surprise you. Not that it will surprise you.