For our purposes, let’s define the “long line” as the set together with a lexicographic ordering:
We want to prove that we cannot have a strictly increasing real-valued function in , meaning that there is no function such that
To start with the proof we are going to require a lemma.
Between two different real numbers there is at least one rational number
The basic idea (lifted from this math.SE answer) is to define “a grid” of rationals that is thick enough to be sure we have at least one rational between the two reals. So, if are the two real numbers, we can use as denominator
and as numerator
Now we need to prove that is always strictly between and .
It’s easy to see that is bigger than ,
and that is less than or equal to :
As we know that is smaller than because
Building an impossible one-to-one function
Using as a name for the previously defined function that takes a pair of reals and returns a rational between them and for an increasing function in , we can define by the following expression:
We know that is one-to-one because if , assuming WLOG that ,
But this result would imply that the cardinality of is smaller than the cardinality of a subset of and that is absurd.
This result would seem a random trivia, but it has an important consequence for microeconomics: lexicographic preferences cannot be described by (real-valued) utility functions.