Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but "ω-consistent". A system is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate "P" such that for every specific natural number "m" the system proves ~"P"("m"), and yet the system also proves that there exists a natural number "n" such that "P"("n"). That is, the system says that a number with property "P" exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. J. Barkley Rosser (1936) strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.