By finite information I mean exclusion of non computable real numbers, random real numbers, etc, or any other mathematical "object" or "element" which would require in general sense an infinite amount of information to define it.
Intuitively it seems to me the answer to the question is no, which would suggest we can always list every element without missing any (bijection with N).
This is trickier than it sounds.
One is tempted to say that, for any notion of "describing" something, the descriptions can be represented as finite objects from a finite alphabet, so there are only countably many descriptions. Each description specifies at most one object, so there are only countably many describable objects.
But this only works if the notion of "describing" is itself definable.
Suppose you decide to say, not unreasonably, that a set is describable if it's first-order definable without parameters. (Note that first-order definability is not itself definable in set theory.)
Then we have:
(1) There is a transitive model of ZFC in which every set is definable without parameters (for example, the minimal transitive model of set theory — the smallest $L_\alpha$ that satisfies ZFC).
(2) There is a transitive model of ZFC in which there are only countably many sets that are definable without parameters (for example, $V_\kappa$ where $\kappa$ is a strongly inaccessible cardinal).
Example (1) above requires the existence of a transitive model of ZFC; this assumption is clearly necessary.
Example (2) above requires the existence of a strongly inaccessible cardinal; this assumption can probably be reduced in strength.
If you believe in a Platonist universe that satisfies ZFC, that's presumably like (2), with only countably many definable sets. But this isn't provable, or even expressible, in ZFC.