6Real numbers
IA Numbers and Sets
6 Real numbers
So far, we have only worked with natural numbers and integers. Unfortunately
the real world often involves rational numbers and even real numbers. In this
chapter, our goal is to study real numbers. To do so, we will first have to define
the real numbers. To do so, we will start from the natural numbers.
Before we start, an important (philosophical) point has to be made. The idea
is to define, say, the “real numbers” as a set with some operations (e.g. addition,
multiplication) that satisfies some particular properties, known as the axioms.
When we do this, there are two questions we can ask — is there actually a set
that satisfies these properties, and is it unique?
The first question can be answered by an explicit construction, i.e. we find
a concrete set that actually satisfies these properties. However, it is important
to note that we perform the construction only to show that it makes sense to
talk about such a set. For example, we will construct a real number as a pair of
subsets of
Q
, but it would be absurd to actually think that a real number “is” a
pair of sets. It’s just that it can be constructed as a pair of sets. You would be
considered insane if you asked if
∃x : x ∈ 3 ∨ x ∈ π
holds, even though it is a valid thing to ask if we view each real number as a set
(and is in fact true).
The next problem of uniqueness is more subtle. Firstly, it is clear that the
constructions themselves aren’t unique — instead of constructing the natural
number 0 as the set
∅
, as we will later do, we could as well define it as
{{∅}}
,
and the whole construction will go through. However, we could still hope that
all possible constructions are “isomorphic” in some way. It turns out this is true
for what we have below, but the proofs are not trivial.
However, while this is nice, it doesn’t really matter. This is since we don’t
care how, say, the real numbers are constructed. When working with them, we
just assume that they satisfy the relevant defining properties. So we can just
choose anything that satisfies the axioms, and use it. The fact that there are
other ones not isomorphic to this is not a problem.
Since we are mostly interested in the natural numbers and the real numbers,
we will provide both the axiomatic description and explicit construction for these.
However, we will only give explicit constructions for the integers and rationals,
and we will not give detailed proofs of why they work.