TITLE: Ordinals - motivation and rigor at the same time
QUESTION [10 upvotes]: Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal with no motivation provided or too simple with no rigor. I've been truly baffled by the concept of ordinals for the last couple days.

REPLY [3 votes]: Starting from the most basic intuition, you use ordinals when you talk about things that are in a certain order: first, second...
This particular order can be represented by a never ending chain with a starting point: for every ring of the chain, there is a next ring that didn't occur before (the chain never loops, goes on forever, and never splits).
So, we could want to find something similar in a set theory, since orders are important for a lot of reasons, the most basic being to be able to say when one ordered set is smaller than another ordered set sharing the way in which it orders its elements. (If we don't care about orders, we can just use cardinality to compare sets)  
In ZFC, we do this: we take a starting point as an ordinal, and decide on a way to generate a "next" ordinal from any ordinal you have.
One way is to take the empty set as a starting point, and define the ordinal that comes after x as xU{x}.
In this way, you have this:
{}
{{}}
{{}, {{}}}
{{}, {{}}, {{},{{}}}}
...
To visualize better, let's name each ordinal after a natural number. We obtain this:
{}=0
{0}=1
{0,1}=2
{0,1,2}=3
...
So, with this method, each ordinal ends up being the set of the ordinals that precede it, if we take the membership relation as the order relation (taking the subset relation works too)
Taking a step further, we could want to expand our horizons and think of an ordinal as anything containing everything that precedes it.
So, let's consider a set containing every ordinal we obtain starting from {} and finding the next one:
ω={0,1,2,...}
There's no problem in considering ω an ordinal too: if we do, it certainly contains every ordinal that precedes it.
In finding ω, we took a 'transfinite step' from the 'finite' ordinals.
Now, we can repeat the process again, by taking ω as a new starting point:
ω+1=ωU{ω}={0,1,2...,ω},
ω+2={0,1,2...,ω,ω+1},
...
And then we can make, again, a transfinite step and go on forever from one level of ordinals to the next.  
We can't, however, have a set of all ordinals, as that leads to Burali-Forti's paradox (very roughly, if there is a set of all ordinals, it would be the last ordinal, but if would also have a following ordinal as any ordinal does, so it can't be the last one).