Visible crew/equipment: When Carrie walks into the bedroom at the start, 2 tape markers are visible on the green carpet. (00:06:55)
Visible crew/equipment: Carrie begs Arthur to move upstairs, when she does, a tape marker is visible on the floor by the stairs to the basement. (00:05:30)
Continuity mistake: Arthur gives Holly a book at the start and she has it in her right hand. Camera cuts and now it is in her left hand. (00:03:20)
Continuity mistake: When the doctor is explaining to Carrie what is wrong with Arthur, he has a stethoscope round his neck that keeps changing position between shots. (00:08:30)
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.