1 a Way Out

We present a way out of Russell's paradox for sets in the form of a direct weakening of the usual inconsistent full comprehension axiom scheme, which, with no additional axioms, interprets ZFC. In fact, the resulting axiomatic theory 1) is a subsystem of ZFC + " there exists arbitrarily large subtle cardinals " , and 2) is mutually interpretable with ZFC + the scheme of subtlety. 1. NEWCOMP. Bertrand Russell [Ru1902] showed that the Fregean scheme of full comprehension is inconsistent. Given the intuitive appeal of full comprehension (for sets), this inconsistency is known as Russell's Paradox (for sets). The modern view is to regard full comprehension (for sets) as misguided, and thereby regard Russell's Paradox (for sets) as a refutation of a misguided idea. We first give an informal presentation of the axiom scheme investigated in this paper. Informally, the full comprehension axiom scheme in the language L(OE) with only the binary relation symbol OE and no equality, is, in the context of set theory, Every virtual set forms a set. We use the term " virtual set " to mean a recipe that is meant to be a set, but may be a " fake set " in the sense that it does not form a set. The recipes considered here are of the form {x: j}, where j is any formula in L(OE). Other authors prefer to use the term " virtual class " , reflecting the idea that {x: j} always forms a class, with the understanding that x ranges over sets. Our terminology reflects the intention to consider only sets, and construct a powerful set existence axiom. We say that {x: j} forms a set if and only if there is a set whose elements are exactly the y such that j. Here y