How do you rigorously define "x<y". The best I can think of is "x<y" equivalent to "y-x positive". But surely "positive" means ">0" and > has snuck in again so we have a circular definition?
How do you rigorously define "x<y". The best I can think of is "x<y" equivalent to "y-x positive". But surely "positive" means ">0" and > has snuck in again so we have a circular definition?
You generally do an axiomatic definition:
A field F is said to be ordered if it satisfies the following axiom:
There is a nonempty subset P of the field F (called the positive subset) for which
(i) If a and b are in P, then a + b is in P (addition closure)
(ii) If a and b are in P, then ab is in P (multiplicative closure)
(iii) For any a in F exactly one of the following holds: a is in P, -a is in P, or a = 0 (trichotomy).
Then you define, for ordered field F and its positive subset P, the < relation as follows:
Let a, b be in F. We say that a < b if b - a is in P. a < b and b > a are equivalent.
So it's a bit axiomatic, usually. There might be an even more rigorous definition in your set theory books. I've given you the definition on pages 16-7 of Kirkwood's An Introduction to Analysis.
Since this was posted in "Analysis, Topology, and Differential Geometry", I think this is appropriate:
Precisely how you define "positive" or "x< y" in the reals depends upon how you define the reals.
One method is the "Dedekind Cut" in which a "cut", A, is defined to be a set of rational numbers satisfying
1) There exist at least one rational number in A.
2) There exist at least one rational number not in A.
3) A has no largest member.
4) If x is in A and y< x, then y is in A.
(Notice that I have used y< x for x and y rational numbers. Since we are defining real numbers in terms of sets of rational numbers we may assume "y< x" has already been defined for rational numbers.)
The set of real numbers is the set of all such Dedekind cuts. That is, a "real number" is such a set.
I won't go into all of the details (which could take several chapters) but we now say that a real number is positive if it is a Dedekind cut containing at least one positive rational number.
Equivalently we say "x< y" if and only if the x's Dedkind cut is a subset of y's.
Another way to define the real numbers in terms of the rational numbers:
Consider the set of all increasing sequences of rational numbers, having an upper bound. We say that two such sequences, $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$, are "equivalent" if and only if the sequence $\displaystyle \{a_n- b_n\}$ converges to 0. It can be shown that this is an "equivalence" relation and so partitions the set of all such sequences into "equivalence classes". The set of real numbers is the set of equivalence classes- a real number is such an equivalence class.
We can now say that a real number is "positive" if and only if for any sequence, $\displaystyle \{a_n\}$, in its equivalence class there exist N such that if n> N, $\displaystyle a_n> 0$. Again, we are assuming that "positive" has already been defined for rational numbers.