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?

- Jul 21st 2010, 12:21 PMJippoRigorous Definition of "Inequality" or "Positive" in the Reals
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?

- Jul 21st 2010, 12:34 PMAckbeet
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*. - Jul 22nd 2010, 12:23 AMHallsofIvy
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.