If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?
For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any $n$ in $\mathbb{N}$, would '$a_1$$a_0$' represent a number?
Is such a system well defined or useful for anything?
On
In a sense, our usual system is like that. In $\LaTeX$ if you put braces around something it gets treated as a single character. I know that isn't what you are thinking, but to do what you are thinking you would need a countably infinite set of characters, which is what we get with the decimal (or other base) system.
If you had them, you could define concatenation to be some operation like multiplication if you wanted.
On
In base 10, we represent a number $n$ as a sequence of digits $n_0, n_1, \ldots$ such that $$n = \sum_{i=0}^\infty n_i 10^i\qquad\text{where } 0\le n_i<10$$
and we require that the sequence of $n_i$ must be eventually zero.
By changing the representation a little bit, we get the so-called factorial base:
$$n = \sum_{i=1}^\infty n_i i!\qquad\text{where } 0\le n_i<i+1$$
and again the sequence $n_i$ must be eventually zero. There is no upper bound on the size of the digits $n_i$.
In this representation, the number 718 is represented as $\langle 0,2, 3,4,5\rangle$ since $$\begin{align}5\cdot 5! + 4\cdot 4! + 3\cdot 3! + 2\cdot 2! + 0\cdot 1! & = \\ 5\cdot120+4\cdot24 + 3\cdot6 + 2\cdot 2 + 0\cdot 1 & =\\ 600 + 96 + 18 + 4 + 0 & = 718.\end{align}$$
This has actual applications; for example it is a useful way to represent a permutation of a list.
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The factorial number system is one such system. Each place value has one more digit than the previous one. It also has the wonderful property that all rational numbers have a terminating factorial system representation.
In general, any mixed-radix system where the number of values represented by each digit is unique is a numeral system with an infinite number of digit values.
The set
$${\mathbb N}[X]$$
is exactly a system as you describe. The constant polynomials are the whole numbers ${\mathbb N}$, and they represent the infinite "symbols" in your system, while $a_0a_1$ is actually the polynomials $a_0+a_1X$.
If you replace $\mathbb N$ by $\mathbb Z$ or $\mathbb Q$ you get some rings which are actually often studied in mathematics.
Added When studying the prime factorization of integers, the same type of system actually comes in play.
Look at the primes $p_1=2,p_2=3,..$.
Then any $n >2$ can be written as $2^{a_1}3^{a_2}5^{a_3}....p_k^{a_k}$ where $p_k$ is the last prime appearing in the prime factorization of $n$.
Then the "symbols" would correspond to the powers of $2$, the elements of teh form $a_0a_1$ correspond to the integers divisible by no other prime than (maybe) 2 and 3 and so on.
Interesting, the example is similar to $\mathbb N[X]$ and the addition of polynomials in $\mathbb N[X]$ corresponds to multiplication in positive integers.