An ancient Greek word (απειρων), pronounced "ah - PAY - rohn," usually translated 'unlimited' or 'boundless', perhaps incalculable. In the Pythagorean "table of the opposites" it appears thus:
limited        apeiron
one many
straight crooked
good bad
nodes freegel
(Unfortunately I don't have an authoritative source for that last pair.)

The Pythagoreans believed that, in the words of Aristotle, "the whole heaven is a harmony and a number" and this gives us a clue to what they thought about apeiron. The ancient Greek concepts of number and harmony were very much bound up in their concept of ratio, in Greek: logos. This has given us words to do with systematic attempts to understand the world: logic, rationality (the same metaphor filtered through Latin.) Apeiron, therefore, can be understood as standing in opposition to this Pythagorean good.

To see how the Pythagoreans might have understood the word, it helps to describe one of the Pythagorean struggles against apeiron. When Pythagoras invented his famous theorem about right-angled triangles, one of the immediate consequences was a demonstration of the existence of what we now call irrational numbers. The Greeks used the terms alogos (inexpressible) and arratos (without a ratio).

Remember, having no decimal point notation, they must conceive fractional parts as ratios between whole numbers. A simple proof shows that the dividend in the ratio for sqrt(2) (the length of the hypoteneuse for a right angled triangle with other sides equal to one, by Pythagoras' theorem) must be both odd and even. In other words, sqrt(2) cannot be exactly represented as a fraction. Even using the modern method, its representation would require an infinite decimal expansion: apeiron.

According to one story, the Pythagoreans took this so seriously that Hippasus, who wanted to go public with the information, was murdered at sea.

Apeiron can also be understood as chaotic, or infinitely complex - the Pythagoreans thought of a crumpled handkerchief as apeiron - and indeed there is a close relationship between the concepts of infinity and complexity.

One way of thinking of the task of defining larger and larger numbers is as the task of packing more and more complexity into a finite definition. But when it comes to the Absolute Infinite, somewhere along the line we give up the ghost.

To my mind, this Pythagorean denial of apeiron is reminiscent of a modern trend in philosophy, which I like to call physicalism. This holds that what is real is what can be treated of by physics. As physics is a formal discipline, this amounts to a denial of apeiron, the undelimited, the informal. This is not to say that physics is wrong, or even incomplete within its own domain, just that the philosophical use of a formalism such as physics as an ontological yardstick is undermined by the implied rejection of the reality of apeiron.

It's quite easy to place the opposing concept of limit, ratio, order, in our world: it's the regular way that stuff behaves when it moves about: Information, causality, natural law. Apeiron might seem harder to spot. But it seems we're immersed in the informal; it's just all the other stuff, the stuff that philosophers get blasted for ignoring! Emotion, values, consciousness, humour, the funny feelings you get when you're drifting off to sleep.. None of these are capable of a realistic treatment by latter-day Pythagoreans, because the anti-apeiron spectacles render them invisible. They must be reconstructed as regularities in behaviour, which, it seems, is precisely what they are not.

See also: Reflection principle.