Why Can't we Describe the Elements in a $\sigma$-algebra?

Why can't we describe the elements of a sigma-algebra? I was told that it is incorrect to say that a given element $B$ in a sigma-algebra $\sigma(A_i)$ can be generated by the $A_i$s using countable set operations.

Can someone provide an example of a sigma-algebra generated by some elements and a set $B$ in the sigma-algebra for which $B$ cannot be expressed in terms of the generators using countable set operations?

• By "describe" I mean write $B$ in terms of the generators of the sigma-algebra using countable set operations. – The Substitute Nov 8 '15 at 10:02