How is "smaller than" defined on $\mathbb{R}$?

Question

According to http://en.wikipedia.org/wiki/Binary_relation

Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science.

AFAIK on the set of real numbers we can define a binary relation $<$ which represents a subset of the cartesian product $\mathbb{R} \times \mathbb{R}$. How do we decide which pairs are contained in this subset? Do we know that by set inclusion? I know about the definition of real numbers with the concept of a Dedekind cut, so I suppose that to know which pairs are in this subset we can view the first element of any pair $(a,b)$ as a set of rationals and if it is included in the second element $b$ than we conclude that the pair $(a,b)$ is contained in the subset of the cartesian product $\mathbb{R} \times \mathbb{R}$ and we can say that $a < b$. Am I right?

Answer 1

If you define the reals with Dedekind cuts, yes:

$$\forall a,b\in\mathbb R(a\leq b\iff a\subseteq b)$$

$$\forall a,b\in\mathbb R(a<b\iff a\subsetneq b)$$

It's a little harder to define ordering on the reals if you use Cauchy sequences. (This is because Dedekind cuts are specifically designed to "complete" the ordering of the rationals, while Cauchy sequences are designed to complete the metric space.)

Answer 2

It all comes down to how we're setting the reals up.

• If we're using Dedekind cuts, then yep, that's how to do it!

• If we're using equivalence classes of Cauchy sequences, then we have to say something messier: $a<b$ if, whenever we pick representative sequences $\alpha\in a$ and $\beta\in b$, there is some $n$ such that for all $m>n$, $\alpha(m)<\beta(m)$. That is, $a<b$ if "$a$'s Cauchy sequences always eventually become smaller than $b$'s Cauchy sequences."

Answer 3

Depends on the construction you make of $\mathbb{R}$. Assuming that you already have built the operators $+,\cdot$, a simple way to notice it would be to make a subset $\mathcal{P}\subset\mathbb{R}$ which represent the positive numbers, then $a\leq b \iff b+-a\in \mathcal{P}$, where $-a$ is the additive opposite of $a$ such that $a+-a = 0$.

Answer 4

In Dedekind's construction $\alpha < \beta$ is defined to mean $\alpha$ is a proper subset of $\beta$. What you say is correct.

Answer 5

If we use dedekind cut to complete Q then then for a < b we must have a $\subset$ b proper subset that is.