Is there any construction of real numbers that does not use a quotient space in the process?

Question

I am working on an isomorphism (in terms of order and operations) from the power set of integers to R. I would like to know if anyone knows of any construction of the real numbers that uses such a simple set to provide it with the structure of the real number system. Thanks!

Answer 1

If we take the rationals as given (or construct them without using a quotient procedure - see my comment below), then the Dedekind cut construction, I think, does what you want.

A real number, the idea goes,$^*$ is completely characterized by the way it "cuts" the number line into two pieces. This cut is completely determined by which rationals wind up on which side, and conversely every "reasonable" partition of the rationals corresponds to a real number (this is the completeness of the reals). This motivates the following definition:

A Dedekind cut is a pair $(A, B)$ of nonempty sets of rationals such that every element of $A$ is less than every element of $B$, $A$ has no greatest element, and $A\cup B=\mathbb{Q}$.

The intuition here is that $(A, B$) is the unique real number "between" $A$ and $B$. E.g. the cut $(A_\pi, B_\pi)$ corresponding to $\pi$ has $3, 3.1, 3.14, ...\in A_\pi$ and $4, 3.2, 3.15, ...\in B_\pi$.

The asymmetry between $A$ and $B$ - where $A$ must have no greatest element, but $B$ is allowed to have a least element - is to accommodate the rationals. The cut corresponding to ${1\over 2}$ needs to put ${1\over 2}$ on one side or the other; we've arbitrarily decided that in this situation we'll put ${1\over 2}$ on the right (the $B$-side).

There is no need to take equivalence classes in this approach: given any real number $r$, there is a unique Dedekind cut corresponding to it, namely $$Cut_r=(\{p\in\mathbb{Q}: p<r\},\{q\in\mathbb{Q}: q\ge r\}).$$ The basic arithmetic structure on the reals "ports over" without serious difficulty; for example, the ordering is given by $$(A_1, B_1)\le (A_2, B_2)\iff A_1\subseteq A_2.$$ That said, the Cauchy sequence construction of the reals allows a much easier development of arithmetic; my point here is just that no serious difficulty crops up with cuts.

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Note that the notion of a Dedekind cut, while geometrically pleasing, is somewhat redundant. We could also just work with the "right side" of a cut:

A half-cut (this is not standard terminology) $C$ is a nonempty upwards-closed proper subset of $\mathbb{Q}$.

The idea is that the real associated to $C$ is its greatest lower bound.

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$^*$Personally, I think Dedekind's original essay does a wonderful job of motivating the idea, and it's still my favorite construction of the reals. The construction of the reals as equivalence classes of Cauchy sequences has lots of advantages over it - in particular, defining the basic arithmetic operations is easier with them than with cuts - but I really find Dedekind cuts to have lots of aesthetic value. And down the road, they wind up being quite important and useful in logic.