A dilemma in the philosophy of set theory (Q1893146)

The author considers a weak form \(\text{BL}_1\) of Bernays' set theory, e.g. determined as NBG with \(\Pi^1_1\)-reflection. Then he shows that \(\text{BL}_1\) proves \(\Delta^1_1 (\text{NBG})\)- comprehension, but that on the other hand \(\Delta^1_1 (\text{NBG})\)-comprehension fails in the case that one assumes that only classes exist which are definable by set-theoretical predicates with set parameters. As \(\text{BL}_1\) is a kind of minimal framework for large cardinals, this means that ``non-predicative'' classes are indispensable for large cardinals.