For every real number x, one of the following is always true:
x < 0
x = 0
x > 0
This is a direct consequence of the real numbers' definition as a complete ordered field (For all x, y, either x<y, x=y, or x>y, so pick y=0).
You would think something as simple and straightforward as this assertion would be uncontroversial, wouldn't you? Definitions of the real numbers such as Dedekind cuts are utterly dependent on the trichotomy law. But even here, we become entangled in the quarrels between various philosophies of mathematics.
Although the trichotomy law for real numbers does not require the Axiom of Choice, it suffers from a similar problem of non-constructibility of a solution for every specific instance. (A broader trichotomy law regarding the comparison of infinite cardinal numbers has been known to be equivalent to the well-ordering theorem, and thus the Axiom of Choice, ever since it was realized such an axiom was needed).
To the Dutch mathematician Luitzen E. J. Brouwer, the Law of Trichotomy was FALSE! Brouwer was the principal force behind the intuitionist movement in the early 20th century. To an intuitionist, if something can't be constructed in a finite number of steps it's gobbledygook.
Pick a normal number of some sort. Brouwer used π, which is probably a bad choice since we don't know if it's normal or not. But there are normal numbers to spare, and for historical reasons we're going to label our chosen number "π", whatever its actual value.
A normal number has the property that any sequence of digits you care to come up with can be found in its decimal expansion. We calculate a new number πˆ (pronounced "pi-hat") using the following method:
- Pick a sequence of digits.
- Find an occurrence of the sequence in the decimal expansion of π
- Examine the first digit in the decimal expansion of π after the chosen sequence. If it's even, add 1. If it's odd,
subtract 1. Exchange 0 with 9 as needed.
- πˆ is the number with the same decimal expansion as π except for this single modified digit.
The real numbers are closed under subtraction, so consider the number πˆ - π. Is it greater than 0, equal to 0, or less than 0?
Decimal expansions are essentially a pseudorandomizing process. If the number we chose is computable† we can plug and crank and eventually wind up with a result. But there are a whole host of normal numbers that are not computable, and for them, there's no guarantee we'll ever find out.
To an intuitionist, this means that πˆ - π is a counterexample to the Law of Trichotomy. There's no way to determine one of the three possibilities and thus none of them hold.
The alternative is to accept that there are true mathematical propositions for which there is no possibility of proof, even if allowed an infinite amount of time. Mathematics would necessarily exist outside the human mind, an apotheosis which is anathema to the intuitionist‡.
On the other hand, a Platonist has no problem with this. It's the foundation of his or her philosophy. You can't prove it? So what? One of the three must be true!
†Brouwer came before Turing and didn't have the concept of computability to work with. But computability is very useful, since it strips away all of the teleological dross to get at the underlying philosophical debate.
‡Ironically, the grandaddy of intuitionists, Leopold Kronecker, was fond of prefacing "All else is the work of man" with "God created the natural numbers". More convincing to me are things like the fine structure constant and the speed of light.
Among other sources, such as an episode of Nova I sketchily recall, is Philip J. Davis and Reuben Hersh's 1981 work The Mathematical Experience. Birkhäuser, reprinted in 1983 by Houghton Mifflin. ISBN 0-395-32131-X
I'd prefer that this be under Law of Trichotomy so you can move it if you want to.
(In case you were wondering: To me, the intuitionist view is inherently solipsistic and I'm not concerned that I'll never know if πˆ - π is less than, equal to, or greater than 0. Brouwer has us chasing a chimera and there are more important things to worry about.)