Throughout some episodes of The Big Bang Theory, there have been some contradictions, which include the following:
Sheldon is sarcastic to Leonard, telling him that "Archimedes would be so proud" in the Pilot episode when Leonard fails after invoking the mathematician's name and teachings. In the next episode "The Fuzzy Boots Corollary", however, it is made apparent several times that Sheldon does not understand sarcasm and even needs a SARCASM sign. It may possibly be that Sheldon knows what sarcasm is, but he can't tell when other people are using it.
Sheldon with the cats.
In "The Fuzzy Boots Corollary", Leonard mentions Sheldon gets asthma when he is around cats, but Sheldon said his family had a pet cat named Lucky and bought 25 cats when Amy "broke up" with him in "The Zazzy Substitution".
In "The Maternal Congruence", it is said that Sheldon cheers for the Grinch's meanness in its movie and finds it a buzz kill when he succumbs to social convention and gives the presents back. In a different episode, Sheldon compares Stuart to "the Grinch in Christmas" in a bad way, showing love for Christmas and hate for the Grinch.
Early in Season 1, Sheldon tells Penny that he is 140 pounds when he goes to the market with her. However, near the end of season 1, he tells his sister to tell their mother that he is 165 pounds, among other things, which would be a 25 pound increase in only a few months.
In "The Luminous Fish Effect", Sheldon estimates that the car he and Penny are riding in is about 4000 pounds. This cannot be correct, as it is a Volkswagen Rabbit (Golf Mk 1) and would be approximately 2000 pounds, maximum.
Howard is described to be 97 pounds and 108 pounds at different points.
Sheldon recalls how he made Leonard wait in line for Star Trek: Nemesis to get good seats, though that film was released in 2002 and Leonard and Sheldon didn't meet until 2003, according to the flashback episodes in "The Staircase Implementation". Given the tepid response to the film by the public at large, it's unlikely there was a revival showing of it, unless it too had unseen footage, that would have made the movie better.
Sheldon is always telling his friends that he does not like touching other people - however, he has made physical contact with Penny (hugging her) and Amy (cuddling), without washing his hands afterwards, or being utterly revolted though he may be getting used to Amy and Penny.
In "The Luminous Fish Effect", Sheldon gave a complicated explanation of what he did with his life past fifth grade, which combined would make him 23 at that time; however, in "The Friendship Contraction", Leonard stated that Sheldon was 30. Given the information in "The Luminous Fish Effect", Sheldon would only be 27.
Amy's name tent does not have "Dr" on it, unlike Raj's, despite both having PhDs.
In "The Porkchop Indeterminacy", Sheldon's sister, Missy is seen as someone who speaks with a heavy Texan accent. However, there are several instances in the episode where she is seen to have dropped her accent.
In "The Love Car Displacement", Amy's name tent did not have the title "Dr" in front of her name, while Sheldon's, Leonard's, Raj's did and obviously Howard didn't. This is clearly a contradiction since Amy had been revealed to have a PhD before this episode. (Then again, it may have been an error on the Lorre and Prady's part.)
In "The Good Guy Fluctuation", Stuart congratulates Leonard for being the first person to pick up a girl in Stuart's comic book store; however, Stuart himself picked up a girl (Penny) in that same comic book store in a Season 2 episode, "The Hofstadter Isotope".
Sheldon intrudes upon Leonard while he is in bed with Joyce Kim. In the "Hamburger Prosulate", when Leonard and Leslie Winkle are having sex, Sheldon is unsure of what to do in those kind of situations, and states that it has never happened before. He asks Penny for advice. But in this episode, he breaks into Leonard's room right when Leonard and Joyce Kim and making out, contradicting Sheldon's statement in "The Hamburger Postulate".
Sheldon also makes a categorical error in "The Bat Jar Conjecture". When Howard says he should answer the engineering questions since he is an engineer, Sheldon compares this to him answering all the anthropology questions because he's a mammal. Anthropology is the study of humans, not mammals; all humans are mammals, but not all mammals are humans. He ought to have said 'mammalogy', or claimed he was a human.
In "The Bad Fish Paradigm", Rajesh admits that he loves trains, indicating that's a passion he and Sheldon share. However in i.e. The Terminator Decoupling Rajesh shows no visible affection to railroad based way of transportation.
In "The Killer Robot Instability" Sheldon and the guys want to improve their fighting robot; however, Sheldon doesn't even know how to open the tool box implying that he has no mechanical skills. groein up back in Texas he did build a cat scanner out of his sister's EasyBake Oven, a sonic death ray and tried to build a nuclear reactor, thus having some skills to invent things.
