Character mistake: Chick Gandil mispronounces Eddie Cicotte's name, calling him "Si-coty" instead of "See-cot."
Character mistake: While pitching his shrinking machine to NASA Szalinski claims that reducing the size of a rocket's payload would result in "staggering" savings in the fuel required at takeoff. However, he makes it clear that his machine works by reducing the space between atoms, meaning that the shrunken object weighs the same as the original. Weight is the critical factor in calculating fuel consumption, not size. Anyone at NASA would pick up on that instantly.
Character mistake: Ray and Egon are giving Oscar the check-up; Ray suggests to run a "Gammill and Pross Infant Acuity Test", which does not exist and is an inside-joke in the screenplay, since Tom Gammill and Max Pross are a duo of comedy writers. Egon agrees and says that they'll finish with an Apgar score, which is a real pediatric term - but it's a test for literal newborns, not an eight months old one like Oscar. (00:16:05)