Research diaries #10: The real line, gaps in the rational numbers and the axiom of completeness

Historically mathematics did not start at the true fundamentals. The fundamentals were only completed around the 19th century. All of the work done before that was done with some intuitive understanding of the basics, in particular the nature of the real line which we will denote by R in this post.

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Mathematicians like Cantor showed that our intuitive understanding of R is flawed. Hence, it requires a rigorous formulation so that we can proceed in a proof theoretical manner to obtain properties of R. Properties of R are of specific interest in relation to the rational numbers which we will denote by Q. Rational numbers are all numbers in R that can be expressed as a fraction of two whole numbers. There exist methods to construct R from Q but that is not the approach we will take as it is rather technical. For the interested you can have a look over here

What kind of fundamental properties can we assume on R? We assume that that we have the natural addition and multiplication so that we can perform all the natural algebraic operations that we did in primary school (more rigorously we are assume that R is a field). We also assume that there exist an ordering on R this means that the elements are ordered but that also the basic operations, addition and multiplication, apply to this ordering. So if a < b then also a+c < b+c (this means that it is an ordered field)

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Although we have not proven the existence of numbers which are not in Q. We expect that there are numbers that are in Q but not in R. These numbers will create gaps in Q in relation to R. So how do we express that R is gapless. We phrase this as an assumption:

Axiom of completeness Every nonempty set of real numbers that is bounded above has a least/smallest upper bound

Here bounded above, also called upper bounded, means that there exists a largest element in R which is greater than all the elements in the set. There can exist many such bounds but the axiom guarantees that there exists a single smallest one. Finally, it important to note that the least upper bound does not necessarily need to be contained in the set. For example if we consider the set generated by the numbers -1/n where n=1,2,3, ... Then 0 is the least upper bound but 0 is not contained in the set.

To get a better understanding of the axiom let's look at it in the context of an example. Consider the set

A = { r ∊ R : r² < 2 }

Then the axiom tells us that is has a least upper bound. However, if we consider

B = { r ∊ Q : r² < 2 }

and imagine that we are only allowed to work on Q then there is no least upper bound in Q. Or more specifically, square root is approximately 1.4142... and consequently can be approximated from above by finer and finer rational approximation meaning 2, 15/10, 142/100 ... but there is no smallest one in Q. It goes on indefinitely. Hence, at least heuristically we observe how the axiom of completeness makes the real line complete

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