> Am I right in assuming that I can not find any such element with my "intuition", but the axiom guarantees that there is a least element?

Well... the second half of that is true. As to the first half, it's not so much that you can't find such an element... you could arbitrarily designate any number in the interval as the "least element", and that would be fine.

Going into it a little more:

Talking about ordering a set, we can define an ordering relation, traditionally called something like <= , to compare two elements. It can be pretty much anything at all; for example, over the set {1,2,3,8} I could define the relation like so, with five parts: 1 <= 1; 2 <= 2; 3 <= 3; 8 <= 8; 2 <= 8. That reflects the idea that they're all equal to themselves, and 2 is less than 8. Any other pair, like 2 and 3, is incomparable under the relation I've defined. In that set, under that relation, the subset {3} has a least element, 3; the subset {2,8} has a least element, 2; and the subset {1,3} doesn't have a least element because 1 and 3 are incomparable.

For a set to be well-ordered, you need some additional properties:

- the order relation must be total, that is, every two elements must be comparable to each other.

- every subset must have a least element under the relation

Using the standard numeric comparison <= , the positive integers are well ordered: any subset, even an infinite one has a least element (though not necessarily a greatest element). The integers are not well ordered under numeric comparison, as they have no least element (-4 is less than -3, but -5 is even less, and so on). But I can easily define a well order "W" on the integers by saying: a W b iff |a| < |b| or (|a| = |b| and a <= b). That W relation produces the order 0 W -1 W 1 W -2 W 2 W -3 ....... , and I can use it to define a least element of any subset of integers.

But you don't have to have an elegant-looking rule to define a well-order. As long as you can choose a least element from any subset, you're good. So if you say "the least element, in my well ordering, of the interval (0,1) is the square root of 1/2", there isn't any problem with that.

However, a constructivist will ask you "what is the least element of the set consisting of all real numbers between 0 and 1, except the square root of 1/2?". And it can be hard to justify the numbers you pick. So I guess human intuition usually doesn't extend to the idea that you can just keep picking seemingly-arbitrary least elements indefinitely (and since the reals are uncountable, you'd actually have to do that an uncountable number of times).

The axiom of choice solves the problem pretty directly; one of its standard formulations is

For any collection of nonempty sets, there is a "choice function" f
which maps every set in the collection to one of its own elements

And so to well-order the reals, you just say the collection of sets will be the set of subsets of the reals, and the least element of any given subset is the value of the choice function f for that subset.

EDIT:

The part in italics above is incorrect (I don't know how to produce strikethrough, or if it can even be done). This always messes me up. In my head, you just repeat the process I described above an impossibly large number of times: least element to be f(R), second least element to be f(R without the least element), etc. But as I mentioned elsewhere, that intuition always leads me into "all sets are countable", which I know is wrong. :(

Anyway, given a subset of R, you can't just apply the choice function; you'd have to start from R and keep applying the choice function and removing values until you got one which was in your subset.