James Pierpont wrote a two-volume work "The Theory of Functions of a Real Variable" which appeared in two volumes. Volume I was published in 1905 and volume II in 1912 Both were reprinted by Dover in 1959.
Volume I page 148 section 242 "The domain of definition of a function of n variables may be any set or aggregate of points in Rn. We with to treat now the most elementary properties of such aggregates which we call point aggregates." ("R" is a "German R" and means the set of all real numbers.)
For the rest of Volume I he uses the term "aggregate"---a cursory search did not turn up another use of "set".
In volume II Pierpont uses the terms "aggregrate" and "set" interchangeably. In some sections of volume II he prefers "aggregate" and in some sections "set". I suspect that the chapters volume II was written or rewritten out of sequence and that sometime between 1905 and 1912 Pierpont got into the habit of using "set", although he did not go back and modify what he had written earlier.
Also a use of the word "class", volume II, page 318, section 289:
"Cantor has divided the ordinal numbers into classes. "Class 1, denoted by Z1, embraces all finite ordinal numbers. "Class 2, denoted by Z2, embraces all transfinite ordinal numbers corresponding to well order enumerable sets; that is, to sets whose cardinal number is Aleph-null"