How can I define the set $\mathbb{R}$ of real numbers?

Question

Assuming that the set $\mathbb{Q}$ of rational numbers is defined, how can I outline the definition set $\mathbb{R}$ of real numbers using the Cauchy sequence?

Thanks :)

Answer 1

Let's start with a concrete example: what is $\pi$?

Well, we know what it isn't: it's not $3$. It's also not $3.1$, $3.14$, or $3.141$.

But those are all good guesses, and what's more, they're improving: $3.1415926$ is closer to $\pi$ than $3$ is. Moreover, they're all rational - that is, we already understand what sort of thing $3.1415926$ is.

This motivates the following "first try":

A real number is a sequence of rational numbers that "ought to converge to something".

Alright, two problems with that.

• What does "ought to converge" mean?

This is where Cauchyness comes into play: the idea is that a sequence of rationals which is Cauchy ought to converge, even if it doesn't converge to a rational - if a Cauchy sequence of rationals doesn't converge to a rational, that means it "points to" a gap in the rationals, that needs to be filled by a (irrational) real.

• Multiple sequences represent the same real number!

This is mathematically more easily dealt with, but maybe conceptually harder to work with. Here's the problem: consider the sequence $$4, 3.2, 3.15, 3.142, 3.1416, ...$$ This sequence also "points at" $\pi$! But it's clearly a different sequence than the sequence $$3, 3.1, 3.14, 3.141, 3.1415, ...$$ that we're more familiar with. The fix is to change our definition to say that a real number isn't a specific sequence, but rather a "collection" of sequences, all pointing the "same way."

This is what ultimately leads us to the right definition:

A real number is an equivalence class of Cauchy sequences.

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Of course, this isn't the full story - in particular, I haven't told you the precise definition of "equivalence" of Cauchy sequences! But hopefully the above conveys the idea reasonably well. Incidentally, you might try to come up with this notion of equivalence yourself - besides being a good exercise, this will also help motivate the definition of Cauchyness.

Answer 2

There are two main ways $\Bbb R$ is constructed form $\Bbb Q$. I'll outline them here:

1. Cauchy sequences. You begin with all possible infinite sequences of rational numbers. Then, for each one, you determine whether it looks like it ought to converge. What I mean by that is that the terms get closer and closer together in a certain, strictly defined, sense (the "Cauchy criterion").

   You throw away the ones that don't satisfy this criterion. Lastly, define $\Bbb R$ as the set of equivalence classes of sequences where two sequences are considered equivalent if the sequence formed by taking the difference of each term converges to $0$.

   Addition and multiplication is defined term-by-term on the sequences, and this carries over nicely to the equivalence classes.

2. Dedekind cuts. You define a real number to be a pair $(A, B)$ of subsets of the rational numbers. In order for such a pair to be a valid Dedekind cut, we demand a few things:

   • Neither $A$ nor $B$ is empty
   • $A\cap B = \emptyset$ and $A\cup B = \Bbb Q$
   • For any $a \in A$ and $b \in B$, we have $a
   • $A$ has no largest element

   In this setting, the real number defined by $(A, B)$ is meant to represent the real number "between" $A$ and $B$, or more specifically, $\sup(A)$. For a Dedekind cut representing a number that happens to be rational, that number is the smallest element in $B$. If the number it represents is irrational, then $B$ has no smallest element. Addition is rather straight-forward; if $(E, F) = (A, B) + (C, D)$, then: $$ E = \{a+c \mid a \in A, c \in C\},\qquad F = \Bbb Q\setminus E $$ Multiplication is defined similarily, but in a more convoluted way, because the product of two negative numbers is positive.