I have been watching that show for a while, but they never mentioned Big's real name.

Big's name is never mentioned. Recall the episode shortly after Carrie met Aiden. She was at the furniture store (gallery, or whatever it was) and Big and Natasha walk up. Carrie introduces Aiden, but when it's time to introduce Big, Carrie spills coffee on him. Similar thing when Big was being introduced to "The New Yankee".

I like the idea of having a character with no name---so mysterious. Its like a bounce off that old ploy of having a character totally talked about but never seen----like Maris on Frasier.

I couldn't believe it the other day when Big left to go to Calafornia.

Someone told me today (probably not true) that in the next season Big gets Carrie pregnant when she visits him in Calafornia.