that the number of elements of [itex]\mathbb{R}[/itex] (seen as a set, obviously) is bigger than the number of elements of [itex]\mathbb{N}[/itex] ...?
Two sets have the same cardinality iff there exists a bijection between the sets. Cantor showed that there is no bijection between [itex]\mathbb{R}[/itex] and [itex]\mathbb{N}[/itex]. A "[URL proof[/URL] of this involves a very simple idea - simple once one has seen it, but not until then.