Incorrect. You CAN use Wifi to connect to the Internet... but not right now.
You see, though Internet LAN capability is possible, it's also somewhat of a toggle. The game itself has to support it for it to work. None of the games currently out support it, so no, you can't play Super Mario 64 DS with people on the Internet. However, when games DO support it (which will be relatively soon unless Nintendo wants fans at their throats), then those games will allow you to play online.
So, to recap, online DS support is something the game developer has to put in their games, not something the DS does by default. And, at the moment, it's not in any of the current games.