Cripples, Bastards, and Broken Things - S1-E4
Corrected entry: When Jon talks to Sam about almost having sex with Ros, he says that their child, in case of a pregnancy, would have been a bastard named Snow as well. This is impossible, because the bastard surnames (like Snow in the North) are only for acknowledged children of noblemen. Jon's children wouldn't have the right to carry a surname, as he's not a nobleman.
Correction: Actually, it is just for noblemen. Bastards of smallfolk (the peasantry) in Westeros do not receive a bastard surname, only those of noble birth do. But Jon is not smallfolk. Despite being a bastard, he is indeed of noble birth and children of noble bastards are allowed to take the surname for themselves and most do.
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