Continuity mistake: When Carrie is having lunch with the lawyers from her office, she is holding a cup and then instantly holding a napkin in the next shot. The cup and napkin switch again depending on the shot. (00:03:10)
Continuity mistake: When Doug and Carrie are arguing about not having gone out for pizza, Carrie has her pen in her hand, but when it cuts it's now on the desk. (00:10:30)
Continuity mistake: When Carrie is sitting at the table talking to Doug about not finishing school, her hand is beside the text book and then instantly holding it from underneath in the next shot. (00:10:35)
Continuity mistake: When Doug says to Carrie that he doesn't even know her anymore, Carrie's laptop and books are open over the desk. However when the camera cuts to her standing up, she has got the laptop folded up and her books in her arms instantly. (00:10:35)
Continuity mistake: In the scene where Carrie and Doug are in the kitchen and Carrie is explaining why she wants to quit school, she fills a glass of water to the top of the glass. A few shots later, she sets the glass down on the counter without taking a drink and it is less than half full.
Answer: To answer the question directly. Carrie and Doug do not have a biological child during the show's main run. In season 3 or 4 finale (cant remember which) Carrie has a miscarriage. Later on in the show they find out Carrie has 1 working ovary. Thus reducing the chance of them conceiving. By the final season, they couple agree to adopt from China and go to collect their child in the show's final episode. While on the flight back from China it's revealed that Carrie is once again pregnant. However the last scene of the show is a flash forward showing that Carrie goes full term with that child and they are struggling with both kids.
Ssiscool ★
Adding on to this answer, I read somewhere that producers were afraid of the show having a "before baby" feeling and then an "after baby" feeling. Some fans feel that main characters, or the show in general, decline in quality after a major change. For example, Jesse and Rebecca on "Full House" after they had twin sons, and Daphne on "Frasier" after she married Niles.