In Modern Greek, the letters omega and omicron are pronounced the same, as a mid-open O vowel. In Ancient Greek, they were different: omega was always a long vowel and omicron was always short (mega = big, micron = small); the capital Ω came from an underlined O (omicron), and the lower-case ω from two o's side by side.

It is clear that omega had a mid-open sound, as in English 'caw': the same noise is indicated onomatopoeically in Greek with omega, for example: κω

They are written in HTML as Ω and ω

In English the name is pronounced with the stress on the O, which is either long (like ohm) or short (omelette). In Greek the -meg- is the accented syllable.

In mathematics, omega has two important uses, one of which I understand well, and the other not at all. They are unrelated. The first one is lower-case; I think the second one is upper-case.

Lower-case omega is the smallest infinite ordinal. It is the order type of the natural numbers in their natural ordering 0, 1, 2, 3, ... . As it is the smallest ordinal of its cardinality, it is equal to the smallest infinite cardinal, aleph-nought (aleph-null).

From it are constructed other ordinals, such as omega+1, omega·2, omega·omega+1, and so forth. The concept and hierarchy of ordinals are well covered elsewhere in E2.

The other Omega is new to me. I've seen references to it in recent years, but it postdates my formal study of maths, so I'm relying on a recent New Scientist article for my rudimentary understanding. (They are strikingly hopeless at expounding mathematics. I never learn anything useful from their maths articles.)

It is a totally random irrational, uncomputable number between 0 and 1, discovered by Gregory Chaitin, and is the probability that a universal Turing machine will halt on a program chosen at random.

The halting problem for a Turing machine can be translated into a diophantine equation. Chaitin constructed a great big diophantine equation in which successive substitutions of N = 1, 2, 3, ... ask whether there are a finite or an infinite number of solutions to the equation, and the truth value of the answer gives the N-th digit of Omega. But because the digits are incalaculably random, so in general are the solutions of most simple equations in number theory. It says here.

If someone knows something about this, an explanation would be appreciated. A /msg to me for very small corrections, but really we need a proper write-up.