Strike Out - S3-E7
Continuity mistake: When Arthur sits down for his meal, there is a bottle on the table. The lid is either on or off depending on the shot. (00:03:30)
Strike One - S3-E5
Visible crew/equipment: When Spence's taxi pulls away, a crew member, light screen and stage light can be seen reflected on the window of the taxi. (00:18:10)
Continuity mistake: When Carrie asks Arthur how a legless dummy will protect her on the subway, her arms are either on the table or off, depending on the shot. (00:10:30)
Soft Touch - S2-E22
Continuity mistake: When Doug is telling Arthur about the filters, there is a leaflet on the table. Between camera cuts, this leaflet is constantly changing position. (00:13:25)
Visible crew/equipment: When Doug walks out of Carrie's room, a boom shadow can be seen just outside the room on the wall. (00:03:35)
Queasy Rider - S2-E1
Continuity mistake: When Carrie says to Doug that he can't have a bike, her hand are behind her back. Camera cuts and now her bag is on her shoulder and her hands are on her bag. (00:03:00)
Queasy Rider - S2-E1
Continuity mistake: When Arthur gets the iron, he has it down by his side. Camera cuts and now it's up in front of his chest. (00:06:30)
Queasy Rider - S2-E1
Continuity mistake: When Doug is checking Arthur's CV at the start, the items on the counter keep moving - noticeably the mayo and lid. (00:00:25)
Queasy Rider - S2-E1
Continuity mistake: When Doug and Carrie are arguing about the smoking for the first time, the cigarette has burnt down nearly half the length in a few seconds. Cigarettes don't burn that fast. (00:12:30)
Queasy Rider - S2-E1
Revealing mistake: Doug falls off the bike at one point and shouts that he has had enough. He opens the door and if you look carefully a stage light and boom mic are reflected in the glass. Slo-mo may help but is not essential for spotting the mistake. (00:15:25)
Continuity mistake: At the start, when Carrie is talking to Arthur about being inseparable she has her jacket over her arm. Camera cuts and now she is putting it on instantly. (00:00:10)
Continuity mistake: When Carrie takes a drink from her cup of tea she is holding her cup with 1 hand in one shot and 2 in a second shot. (00:14:30)
Queasy Rider - S2-E1
Continuity mistake: As Doug is telling Carrie he hates the bike, his beer and crisps vanish instantly from the sofa. (00:19:20)
Hi Def-Jam - S3-E13
Visible crew/equipment: When Mr. Purzan finds out that Stephanie doesn't want to date him and runs off like a little nancy-boy, we see his reflection pass through the picture in the hall. A moment later we can see the reflection of two people pass in the opposite direction but no one walks back into the room. (00:20:30)
Strike Out - S3-E7
Continuity mistake: After Doug says "Let's hit the streets", Doug, Arthur, Deacon and Kirby go on a mini crime-spree. During this montage each characters outfit completely changes several times. They are also dressed very lightly for late October/early November in New York. (00:13:00 - 00:14:00)
Continuity mistake: During the first shuffleboard scene, the characters are brightly lit and cast by long dark shadows (indicating early morning or late afternoon), but when you see shots of the shuffle pucks flying into the bleachers, the bleachers' and extras' shadows are cast straight down (indicating noon hour).
Continuity mistake: When Arthur asks Doug for a game of ping pong, a ball suddenly appears under his bat and then disappears and Doug throws him the ball to have the first serve.
Visible crew/equipment: When Doug mentions about a mistake in the word jumble, look in the glass above him to see a boom mic reflection clear as day. (00:01:45)
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.