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Existence of some sparse sets of nonstandard natural numbers

Published online by Cambridge University Press:  12 March 2014

Renling Jin
Renling Jin*
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA, E-mail: [email protected]
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Abstract

Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. 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 Є ℋ ∖ {0}. In §2, 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 , (ii) there does not exist any U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided . We obtain some partial results for the case U = V.

Keywords

hyperfinite integercutcountably determined setU-discrete set
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 959 - 973
Copyright
Copyright © Association for Symbolic Logic 2001

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References

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