Can I use greek letters such as alpha to denote a set? Can we have a set which is called by a small greek letter, e.g. a set $\alpha$?
 A: It's perfectly "legal" to do so, and in set theory lowercase Greek letters are used as variables for certain kinds of sets (namely, ordinals). (And sometimes they are used as reals, but generally for constant values, coefficients, and the like, much less commonly for arbitrary reals.) It's best to stick to common conventions – and there are several of those. 
One convention is that, within a theorem or a proof (or a paper or a book), consecutive letters from the same alphabet generally denote objects of the same kind. Each area of mathematics has its own conventions, preferences, and traditions. Here are a few:


*

*$i, j, k, l$ – integers, often indexes, subscripts and the like

*$m, n$ – integers

*$p, q, r$ – rationals

*$x, y$ – anything

*$f, g, h$ – functions

*uppercase $A, B, C, ..., M, N, .., X, Y, Z$ – "bigger" things, such as sets, matrices

*$G, H$ – groups

*fancier names like $\mathscr{A}, \mathscr{B}, \mathscr{C}$ – families of of sets or the like: collections of things which contain things, so, at least "2 levels up" from the basic entities under discussion

*(set theory) $\alpha, \beta, \gamma$ – ordinals

*(set theory) – $\kappa, \lambda$ – cardinals


"Bigger" glyphs generally correspond to "bigger" structures. 
Consistency counts for a lot. Variable names that respect conventions tell you at a glance the type of the object, so a reader (who might be you in the future) doesn't have to puzzle over "what the heck is '$\mathscr{A}$', again?... Oh right, a positive real between $\mathfrak{s}$ and $\Theta$. <muttered curses>". You're writing to be read and understood, after all, and math is hard enough already that your readers don't need additional burdens imposed by random names.
A: As far as I am concerned,
you can use anything you want.
However,
I think that sets are
usually denoted by capital letters,
and their elements by lower case letters.
A: As some other answers have indicated, (a) there are no rules against denoting a set using a lower-case Greek letter, and (b) there are conventions which use other kinds of letters (typically capital letters for sets).
As a result, you should have a clear reason for why you are contravening those conventions.  Yes, they are only conventions, and may be violated when the situation warrants it.  Some examples of when the situation warrants it:


*

*The sets are generalizations of ordinary single-valued variables.

*Capital letters have already been consistently used for something else.


*

*They may, for instance, be sets of a different kind; we might have $\alpha = \{A, B, C, \ldots\}$.


*The sets are operated on in ways that resemble arithmetic.


Obviously, this list isn't exhaustive, merely suggestive.  If you have a good enough reason to name a set a lower-case Greek letter, go ahead and do it, so long as you are clear on your notation.
A: You can use any symbol you want. Unconventional symbols should probably be defined in order that the reader not be confused.
For example, it is perfectly valid to use the symbol "$\leq$" to denote a the set whose members are the integers $2$, $5$, and $7$. Then I could write statements such as $$7\in\leq$$ and $$\leq\subset\mathbb Z$$
But unless I had explicitly made the meaning of the symbol clear, the reader wouldn't have the slightest idea what I meant and would probably think it was a typo.
Your proposed symbol for a set, however is perfectly fine. Convention might suggest using capital Latin letters ($A,B,C,\ldots$), but in context I don't see why "$\alpha$" would ever be misunderstood.
