Julia Robinson numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1565-1599
Author(s):  
Pierre Gillibert ◽  
Gabriele Ranieri
Keyword(s):  
Infinite Family ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
First Order Theory ◽  
Infinite Degree ◽  
Totally Real

We construct an infinite family of totally real algebraic extensions of [Formula: see text] whose ring of integers has a Julia Robinson number distinct from [Formula: see text] and [Formula: see text]. In fact, the set of Julia Robinson numbers obtained is unbounded. This gives new examples of algebraic extensions of [Formula: see text] of infinite degree whose ring of integers has undecidable first-order theory.

LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

2011 ◽  
Vol 22 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
MICHEL RIGO ◽  
LAURENT WAXWEILER
Keyword(s):  
Finite Field ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
Polynomial Coefficients ◽  
Bounded Number ◽  
First Order Theory ◽  
Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

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