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A Q set is an uncountable set X of the real line such that every subset of X is an F" relative to X. It is known that die existence of a Q set is independent of and consistent with the usual axioms of set theory. We show that one cannot prove, using the usual axioms of set theory: 1. If X is a Q set men any set of reals of cardinality less than the cardinality of X is a Q set. 2. The union of a Q set and a countable set is a Q set. The existence of a Q set is a fundamental question of setdoi:10.1090/s0002-9939-1980-0550513-4 fatcat:7jgqwzsfmzgupfgcwiaqhqkk4a