Revealing mistake: When Carrie phones Doug at IPS, Doug goes to press the button on the phone for the correct line, but doesn't press any and continues to talk to Carrie. (00:07:40)
Revealing mistake: When Arthur is looking in Carrie's bag, there is one shot with a blue kettle next to the bag. 10 stage lights can be seen reflected in the kettle. (00:11:50)
Revealing mistake: Just as Arthur goes to look in Carrie bag, there is a wide shot and a cabinet can be seen by the door. In the glass of this cabinet's door, you can see a stage light reflected. (00:11:45)
Revealing mistake: When Doug tells Arthur his toast is done, there is no smoke, but as soon as he is finished speaking there is smoke coming from the toaster. (00:04:05)
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.