Frege's Full Comprehension Scheme in Modern Mathematics

Frege’s full comprehension scheme, formulated in "Grundgesetze der Arithmetik", was central to his logicist project. In its unrestricted form, however, it led directly to Russell’s paradox and was therefore rejected as a foundation for mathematics. This chapter revisits Frege’s comprehension principle from the perspective of modern set theory. Drawing on historical analysis and contemporary mathematical practice, I argue that although full comprehension cannot be sustained as a formal axiom, an informal version continues to guide set-theoretic reasoning. By examining the blurred boundary between sets and proper classes in current mathematical work, I suggest that Frege’s original idea survives as an informal working principle, applied competently by practitioners without generating contradictions.