We look at infinite levels of the Ershov hierarchy in the natural system of notation, which are proper for jumps of sets. It is proved that proper infinite levels for jumps are confined to Δ −1 a -levels, where a stands for an ordinal ω n > 1.A set A is said to be n-computably enumerable (n-c.e.), where n ∈ ω, if there exists a computable function f in two variables such that for any x, we haveClasses of n-c.e. sets, where n ∈ ω, are initial levels of the Ershov hierarchy [1-3]. For ordinals greater than or equal to ω 2 , appropriate classes depend on the notation for these ordinals. In the present paper, we opt for the following natural system of notation for ordinals less than ω ω :where a is a number of a row k 0 , . . . , k m of length m + 1.In [1][2][3], it was shown that each level of the Ershov hierarchy contains sets not belonging to lower-lying levels. Here we describe levels containing Turing jumps missing at lower-lying levels of the Ershov hierarchy. It is not hard to show that not each such level properly contains the Turing jump of a set. Thus, for instance, there do not exist a set A and a number n > 1 such that A would be properly n-c.e. In fact, if a jump A of a set A is n-c.e. then A is also n-c.e. If A is computable then A is c.e. Suppose A is not computable. Denote by B a set such that ∅ < T B T A and B is c.e. By using the resolution method, we can construct a non n-c.e. set C computable with respect to B. In this event C is 1-1 reducible to A , which is a contradiction *

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