Jeannie says that her place of birth is Babylon. But in the earlier episode "G.I. Jeannie" she states that Pompei was her place of birth.
Tony tells Jeannie she probably knew Rome under the name Gaul. However, Gaul was in fact the Roman name for France.
Major Nelson stated that he's going by steamship to Europe. Although steamships between the United States and Europe still existed in 1966 most people at this point were flying by plane to Europe. This would especially be true if a Air Force pilot and NASA astronaut such as Major Nelson. Within a few years of this episode most of the remaining steamships would be retired.
When Roger shows Anthony the cage with the parakeet in the first scene the parakeet is at the bottom of the cage but in the next scene the parakeet is hanging on the center of the cage moving.