Rudin Analysis: order in set of rationals In Rudin's Analysis, chapter 1, page 3, definition 1.6 states

$\mathbb Q$ is an ordered set if $r<s$ is defined to mean that $s-r$ is a positive rational number.

But the very definition of 'positive' requires the notion of greater than or less than, so it seems all circular to me.
 A: Original Answer No.  We have what a "positive rational number" means (it is the quotient of a positive integer by another positive integer), but we don't yet have the ordering $<\subseteq\mathbb{Q}^2$.  This definition defines the ordering on $\mathbb{Q}$.

Addendum OK, checking my copy of baby Rudin.  Definition 1.6 is

1.6 Definition An ordered set is a set $S$ in which an order is defined.

What you quoted isn't Definition 1.6, but an example in the paragraph after Definition 1.6.  Anyway, the circularity is avoided if you do it properly as suggested in my original answer.
A: You can define a positive rational number with an algebraic property : it is a rational number which is a non zero square of a real number. That is why he is doing this. 
EDIT : sorry, it is not a natural way to do it, since we construct $\mathbb{R}$ after $\mathbb{Q}$. So, the natural way is to say that a rational $p/q = (p,q)$ (rational defined by a equivalence relation on 2-uplets) is positive if $p$ and $q$ are positive, or $p$ and $q$ are negative.
A: The positive rational numbers can be defined without reference to order by


*

*$1\in \mathbb Q^+$

*$a,b\in \mathbb Q^+ \rightarrow a+b\in\mathbb Q^+\land ab^{-1}\in\mathbb Q^+$

*$a\in \mathbb Q^+ \rightarrow -a\notin \mathbb Q^+$
1 and 2 ensure $\mathbb Q^+$ contains all positive rationals: any positive rational can be written as the ratio of two positive integers, which can themselves be written as finite sums of $1$s. 3 then ensures that no negative rationals or $0$ can be elements of $\mathbb Q^+$.
A: I have an analysis textbook which lays out axioms for the Reals, in which the term 'positive' is an undefined term satisfying the following axioms:
1) For any Real number r, one of the following mutually-exclusive conditions holds: (a) r is positive; (b) r is zero; (c) -r is positive.
2) Any finite sum of positive Real numbers is positive, and any finite product of positive Real numbers is positive.
From these it is not too hard to prove basic properties of positive numbers, e.g: 1 is positive; -1 is NOT positive; N is positive for each natural number N; etc.  Then the ordering r < s (for Real r, s) is DEFINED by '(s - r) is positive'.  No circularity.
