TAME EXPANSIONS - PROBLEMS
Problem 1. Consider the structure (R, <, +, −, 0, 1, (fq )q∈Q , λ), where fq : R → R
maps x to qx and λ : R → R denotes the greatest integer function on R, that
is λ(x) = max(∞, x] ∩ Z. Show that the theory of this structure has quantifier
elimination.
Problem 2. Show that every Borel sets is definable in (R, <, +, ·, Z).
Problem 3. Show that the theory of (R, <, +, −, 0, 1, Q) admits quantifier elimination. What happens if you replace Q by a Hamel basis (ie a basis for the Q-vector
space R)?
Problem 4. (Open) Let fr : R → R maps x to rx. Let r ∈ R \ Q. What can you
say about the definable sets in (R, <, +, −, 0, 1, fr , Q)?
Problem 5. (Open) Let R be an expansion of a ordered field such that every
bounded definable subset of R has a supremum. Let X ⊆ R be a definable set that
is closed, bounded and discrete, and let f : X → X be definable and injective. Is f
surjective?
Date: July 12, 2016.
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