In the preface of Fundamentals of Abstract Analysis by Andrew M. Gleason, he writes,
The whole book is based on naive set theory... On the other hand, I believe that every theorem and proof will withstand a translation into the Hilbert-Bernays-Gödel axiomatic system.
I can't find anything in Google for Hilbert-Bernays-Gödel, so I imagine it might be an alternative name for something (the preface was written in 1966, so terminology may have changed). Perhaps Von Neumann–Bernays–Gödel set theory? Does anyone knowledgeable in foundations know what Gleason means by "the Hilbert-Bernays-Gödel axiomatic system"?