TITLE: Indeterminate "$x$" in algebra/ring Theory
QUESTION [6 upvotes]: How do you interpret the indeterminate "$x$" in ring theory from the set theory viewpoint? How do you write down $R[x]$ as a set? Is it appropriate/correct to just say that
$$R[x] = \{ f: R \to R \mid \exists n \in \mathbb{N}, \mbox{ and } c_0, \dots, c_n \in R \mbox{ s.t. } f(x) = c_0 + c_1 x + \dots + c_n x^n \}$$
This appears to be a very analytic definition. Is there a better definition that highlights the algebraic aspect of the set of polynomials? $$ $$
EDIT: There is a thread on this at http://mathoverflow.tqft.net/discussion/568/inconsistent-and-closedminded-question-closing/#Item_0 
There is a single user who is composing a lengthy response to this question. Anyway, take a look.

REPLY [3 votes]: This began as a comment on Anweshi's answer, but it got too long.  
I consider the functorial definition (as in Anweshi's answer but slightly modified --- see below) to be the best one, because it tells us what is really important about $R[x]$.  Other definitions (using sequences, modules with products, symbols, etc.) are better viewed as constructions rather than definitions; they provide proofs that the functorial definition can be satisfied.  The situation is analogous to the case of the real numbers, whose important property is (and whose definition therefore should be) that they form a complete Archimedean ordered field; Cauchy sequences, Dedekind cuts, etc. provide constructions.  
The slight modification that I'd make in Anweshi's formulation of the functorial definition is that I regard this definition not as simply defining $R[x]$ but as defining the pair $(R[x],x)$.  The most obvious reason for the change is that $x$ is explicitly used in the statement of the universal property.  In fact, the definition simply says that the pair $(R[x],x)$ is the universal example of a ring with a homomorphism from R and a specified element.  (I've been assuming that we're dealing with commutative rings; otherwise, I should either say "a specified central element" or work with fancier "polynomials" where the coefficients and $x$ can be interleaved in monomials, rather than just having a coefficient times a power of $x$.)  A side benefit of this form of the definition is that it gives an immediate answer to the question "what is $x$": it is the specified element in the universal example.