Wasn't TWA founded in 1925? I believe it's either TWA or Western Airlines (which, at one time was part of TWA [Transcontinental & Western Air Express]). So, depending on which branch of the family tree you look at, it's probably one of those carriers.
And since Western is now part of Delta, do we consider Delta the oldest carrier?
"If I don't manage to fly, someone else will. The spirit wants only for there to be flying. As for who happens to do it, in that he has only a passing interest."
- R.M. Rilke