Existence of Some Sparse Sets of Nonstandard Natural Numbers

Answers are given to two questions concerning the existence of some sparse subsets of H = f0; 1; : : : ;H 1g N, where H is a hyper nite integer. In x1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set X H which contains exactly one element in each U -monad, if and only if U = a N for some a 2 H r f0g. In x2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V H such that for any cut U , (i) there exists a U -discrete set X H with X +X = H (mod H) provided U $ V , (ii) there does not exist any U -discrete set X H with X+X = H (mod H) provided U % V . We obtain some partial results for the case U = V . 0 Notation and De nition We always work within a xed !1-saturated nonstandard universe V in the sense of [1]. The reader is assumed to be familiar with the basic de nitions and facts about nonstandard universe and nonstandard analysis. Those de nitions and facts could be found in x4.4 of [1], [2] or [7]. We write ; ; : : : for ordinals and ; ; : : : for cardinals. We write ! for the rst in nite ordinal which is also the set of all standard natural numbers. For any n 6 ! we write 2 for the set of all functions from fk 2 ! : k < ng to 2. We write also 2 for the set S n2! 2. Let N denote the set of all natural numbers and Z denote the set of all integers in V . For any real number r in V , let [r] denote the greatest integer in Z which is less than or equal to r. For any set S, let card(S) denote the set theoretic cardinality of S. If S is an internal set, then let jSj denote the internal cardinality of S in V . For any a; b 2 Z we write [a; b] exclusively for the interval of integers, i.e. [a; b] = fx 2 Z : a 6 x 6 bg. Similarly for [a; b), (a; b] and (a; b). In the cases of [a; b), (a; b] or (a; b) we allow the numbers at the open-ends being non-integer real numbers. For example, ( 5 2 ; 5 2 ) = f 2; 1; 0; 1; 2g. Mathematics Subject Classi cation Primary 03H05, Secondary 03E15, 11P99