2006/09/01 by David Marker · 4 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematics #Countable set #Predicate (mathematical logic) #Model theory #Complement (music) #Combinatorics #Corollary #Discrete mathematics #Order (exchange)
paper · doi:10.2178/jsl/1154698577
openalex publication_date 2006/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
When studying the model theory of the first observation is that the integers can be defined as Since ∂ exp is subject to all of Gödel's phenomena, this is often also the last observation. After Wilkie proved that ℝ exp is model complete, one could ask the same question for ∂ exp , but the answer is negative. P roposition 1.1. ∂ exp is not model complete P roof . If ∂ exp is model complete, then every definable set is a projection of a closed set. Since ∂ is locally compact, every definable set is F σ . The same is true for the complement, so every definable set is also G δ . But, since ℤ is definable, ℚ is definable and a standard corollary of the Baire Category Theorem tells us that ℚ is not G δ . Still, there are several interesting open questions about ∂ exp . • Is ℝ definable in ∂ exp ? • (quasiminimality) Is every definable set countable or co-countable? (Note that this is true in the structure (∂, ℤ, +, ·) where we add a predicate for ℤ). • (Mycielski) Is there an automorphism of ∂ exp other than the identity and complex conjugation? 1 A positive answer to the first question would tell us that ∂ exp is essentially second order arithmetic, while a positive answer to the second would say that integers are really the only obstruction to a reasonable theory of definable sets. A fascinating, novel approach to ∂ exp is provided by Zilber's [6] pseudoexponentiation. Let L be the language +, · E , 0, 1.