Hey,
Could anyone give me a litte explanation of what ordinals are ? (Well, especially infinite ordinals)
I understand what ordinals are for finite sets, but not really for infinite sets.
The Wikipedia article basically treats all the infinite ordinals in a quite formal way, saying is the order type of (all right then), and then say that is the order type of , then is the order type of ... And then comes , then later on , , ... then ,..., then , and after that sequence of powers (and the other ordinals between each successive power) comes (this is where I started to get confused, but ok I'll accept that as a definition). Then we got all the same business with (with the powers and all), but then what comes after ? (By the way, how do you do \ddots (that's ) in the other way ?) Or is there just no definition because it's quite irrelevant (we could make up a letter, then we could make up another, and another, and another again...)
Then they say there's an that's the set of all countable ordinals (under the understanding that we identify each ordinal with the set of ordinals less than that ordinal), and obviously this means is the smallest uncountable ordinal. But what does this ordinal effectively mean ? All we've done is formally define a succession of ordinals and arrived at the conclusion that there must be uncoutable ordinal numbers.
So I'm quite confused, I find the wikipedia article excessively formal, what do the ordinals effectively mean ?
(I'm not even going to think about infinite cardinal numbers indexed by infinite ordinals)

Sam

Ordinals are basically a very powerful set theoretic tool for counting, comparing and ordering. I find that an axiomatic approach to them based on Von Neumann's definition can be easier to handle and understand than the usual one with order classes. You basically need just two axioms (not in the sense of the axioms of ZFC but rather in the sense of the group axioms): is an ordinal if and only if (1) implies and (2) for all precisely one of and holds ( is totally ordered by ). You can then, just from these two simple properties (and ZFC, obviously), prove a lot about the ordinals and get to very powerful results.

For starters, note that is an ordinal (the first one) and also for any ordinal the "successor" is also an ordinal (check it using the definiton). This actually constructs all the finite ordinals, i.e. the non-negative integers, by taking and for any (this is the usual Von Neumann definition of the non-negative integers). It's quite easy to prove that for any two ordinals and , one of and holds, such that the ordinals are totally ordered by (even stronger, any set of ordinals is well ordered) and we can also prove that every set of ordinals has a supremum according to . Moreover, an ordinal turns out to always be the set containing all smaller ordinals, which, for example, allows us to characterize ordinals that aren't the successor of any other ordinal (called limit ordinals and distinguished from successor ordinals): it's the case of (which basically represents ), the next biggest ordinal after the finite ones (their supremum), i.e. (feel free to check this is a well defined ordinal). From there on, you can continue this process to generate bigger and bigger ordinals like the ones you mentionned, i.e. , , and so on to ordinals like and and so forth (I don't think they have any specific names after that, but has one because it's rather special and still useful - look it up on Wikipedia). Accordingly, is simply the supremum of (or, again, "set of") all countable ordinals.

Fast forward to some really interesting stuff: as it happens, using the Axiom of Choice, you can prove that for any set , there is a bijection between and some ordinal . This is immensely powerful: for instance, defining an order on as if and only if in grants us a well-ordering of any set (this is known as the well ordering principle and is yet another counterintuitive statement equivalent to the Axiom of Choice). It's also what allows us to define the cardinal of a set : it's meerly the smallest ordinal (according to , as usual) for which there is a bijection to (this exists since by the theorem there is at least one such ordinal and any set of ordinals is well ordered).

Hopefully this gives some idea of what ordinals are and what they're useful for. I'm fully aware of how hopelessely unclear some of my explanations are, so do ask questions if you need any sort of clarification :P.

Thanks a lot, that's immensely helpful.
Just one question :
When you say that this well ordering principle allows us to define the cardinal of a set as the smallest ordinal for which there is a bijection to , doesn't this mean ? Shouldn't this be ?

Otherwise, thanks a lot, it was very clear.

Well, I wasn't actually talking about the well ordering principle itself, just the theorem that there is a bijection between any set and some ordinal (technically speaking though, that's equivalent). Even so, yes, both and are actually true statements: in fact, the first one is true because of the second one and because of the fact that there's no smaller with . So, we actually define as , but we still use different names because of the connotations attached to each. All cardinals are in this sense ordinals, that's what allows us to define them properly. Sticking to this notation, you also have stuff like , and so on.

Dammit cosmin, why'dyou have to suck all of the banter away with such a spot on explanation...

Wrong forum i suppose.

I could have also pointed out that I have a proof by (transfinite) induction that every ordinal is the null set, which effectively solves the Generalized Riemann Hypothesis, gives elementary proofs of Fermat's Last Theorem, the irrationality of (and of bad proofs by induction), the Continuum Hypothesis (it turns out it was decidable and true after all), Kenya's food and health problems, Goldbach's conjecture, a general unified field theory (where every particle is seen as a tiny piece of manx oak smoked cheddar, thus proving that "MANX RULE OVER ALL"), a recipie for the ultimate cookies, a method for constructing odd perfect numbers, the 3n+1 problem, Danny Carey is god, the inconsistency of Mathematics, the Twin Prime conjecture, All Your Base Are Belong To Us, a stronger version of P=NP (everything is effectively P), (thus eliminating the "stupid and evil singularity problem") and, last but not least, the logical consistency of "It's time to kick ass and chew bubblegum and I'm all out of gum". Surely that would have killed all the mathematical banter of the universe!

Doesn't help you find the beef though.

Actually it does, it proves the beef is everywhere (and in fact everything IS beef).

.

Did you really need to use Tom, a little overkill for an accent?

Café

I learnt how to do accents so I could spell Pokémon right

(It's Alt Gr and e)

Ŧħåťŝ řǽłļÿ ñôŧ ŝœmèţĥîñg ýõû şħôǚŀđ ãďmïţ ŧō. :P

Sτοπ χοωινγ οφφ !
Aκκεντσ αρεν'τ σομεθινγ το βε προυδ οφ. (Aνδ νοτ θισ ειθερ)
(Nο χεατινγ)

Aνιωαι, ζισ κειβοαρδ δοεσ ιτ αλλ. Yου δον'τ τακε ρισξ. θισ εκλιψεσ ιου.

(Uσεδ αλλ λεττερσ, ηει !)

(Υεσ, θισ ποστ ισ υνκουνταβλε)

I'm worried that I can actually understand that without a greek alphabet table.

Well the Greek Alphabet is a useful skill to have in Greece, even if you don't speak the language. Looking for road signs "What did that Greek one say?" "Dunno, wait for the English one" "Yes it's this... it was that turning"