Revealing mistake: As Carrie skids into the parking spot at the start, skid marks are visible on the floor from previous takes. (00:00:30)
Continuity mistake: In the overhead shot when Carrie parks there is no-one around. From the internal shot, there are 2 people visible through the side windows. (00:00:40)
Continuity mistake: Carrie and Arthur are talking about Carrie making a Thanksgiving meal, and as they are talking the frozen meals in Carrie's trolley change position between shots. (00:03:10)
Continuity mistake: Just after picking the big turkey, Carrie's bag keeps changing place in the trolley. Note its position in relation to the handle. (00:06:00)
Visible crew/equipment: After Arthur coerces Carrie into inviting quite a few others to their dinner she reminds him of his inside voice, then, when Arthur walks away to get the gherkins in aisle 4, there are multiple visible T-marks taped to the floor behind Carrie. (00:07:05)
Continuity mistake: When Carrie asks Doug what he's been doing, her hands change place instantly. (00:07:20)
Continuity mistake: When the guy chucks his cheque book into the basket, it reappears in his hand a second later. (00:11:20)
Visible crew/equipment: When Arthur is talking about the store clerk hiding yam's out the back, there is an orange marker on the floor between the two men. (00:14:20)
Visible crew/equipment: When the store clerk walks off from talking to Arthur an orange T marker is visible where he was stood. (00:14:50)
Visible crew/equipment: Doug calls the phone number of the mystery guy, and when mystery guy answers his cell just a few feet away from Doug, two green tape marks are visible on the floor near Doug's feet. (00:16:30)
Visible crew/equipment: While Doug, Carrie, and Arthur are on the checkout line, Doug notices the mystery guy on another line, and two tape marks (an orange and a green) are visible on the floor near Doug's shopping cart. (00:20:20)
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.