scholarly journals An Invitation to Model-Theoretic Galois Theory

2010 ◽  
Vol 16 (2) ◽  
pp. 261-269 ◽  
Author(s):  
Alice Medvedev ◽  
Ramin Takloo-Bighash
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Order Theory ◽  
Closed Field ◽  
Finite Sets ◽  
Algebraically Closed Field ◽  
First Order ◽  
First Order Theory ◽  
Special Case ◽  
Large Model

AbstractWe carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.

Sur la théorie élémentaire des corps de fonctions

1986 ◽  
Vol 51 (4) ◽  
pp. 948-956 ◽  
Author(s):  
Jean-Louis Duret
Keyword(s):  
Algebraic Geometry ◽  
Function Fields ◽  
Order Theory ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
First Order ◽  
First Order Theory ◽  
Nonzero Characteristic

AbstractWe study the first order theory of function fields in the language of fields by using fundamental results on curves in algebraic geometry. We give some applications; for example, using a theorem of G. Cherlin, we prove the undecidability of function fields with nonzero characteristic over an algebraically closed field.

Properties of forking in ω-free pseudo-algebraically closed fields

2002 ◽  
Vol 67 (3) ◽  
pp. 957-996 ◽  
Author(s):  
Zoé Chatzidakis
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Order Theory ◽  
Absolute Galois Group ◽  
Prime Field ◽  
First Order ◽  
First Order Theory ◽  
Closed Fields ◽  
The Absolute ◽  
Algebraically Closed Fields

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

Decision problems concerning properties of finite sets of equations

1986 ◽  
Vol 51 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Cornelia Kalfa
Keyword(s):  
Order Theory ◽  
List Type ◽  
Finite Sets ◽  
Operation Symbol ◽  
First Order ◽  
First Order Theory ◽  
Rank One ◽  
Joint Embedding ◽  
Algebraic Language ◽  
Joint Embedding Property

In this paper a general method of proving the undecidability of a property P, for finite sets Σ of equations of a countable algebraic language, is presented. The method is subsequently applied to establish the undecidability of the following properties, in almost all nontrivial such languages:1. The first-order theory generated by the infinite models of Σ is complete.2. The first-order theory generated by the infinite models of Σ is model-complete.3. Σ has the joint-embedding property.4. The first-order theory generated by the models of Σ with more than one element has the joint-embedding property.5. The first-order theory generated by the infinite models of Σ has the joint-embedding property.A countable algebraic language ℒ is a first-order language with equality, with countably many nonlogical symbols but without relation symbols, ℒ is trivial if it has at most one operation symbol, and this is of rank one. Otherwise, ℒ is nontrivial. An ℒ-equation is a sentence of the form , where φ and ψ are ℒ-terms. The set of ℒ-equations is denoted by Eqℒ. A set of sentences is said to have the joint-embedding property if any two models of it are embeddable in a third model of it.If P is a property of sets of ℒ-equations, the decision problem of P for finite sets of ℒ-equations is the problem of the existence or not of an algorithm for deciding whether, given a finite Σ ⊂ Eqℒ, Σ has P or not.

Partially ordered interpretations

1977 ◽  
Vol 42 (1) ◽  
pp. 83-93
Author(s):  
Nobuyoshi Motohashi
Keyword(s):  
Basic Result ◽  
Predicate Symbol ◽  
Order Theory ◽  
Free Variable ◽  
Consistent Theory ◽  
First Order ◽  
Partially Ordered ◽  
First Order Theory ◽  
Order Language ◽  
Special Case

In this paper, we shall define the “partially ordered interpretation” of a first order theory in another first order theory and state some recent results. Although an exact definition will be given in §4 below, we now give a brief outline. First of all, let us recall the “interpretations” defined by A. Tarski et al. in [17] and the “parametrical interpretations” defined by P. Hájek in [6], [7] and U. Felgner in [3]. Since “interpretations” can be considered as a special case of “parametrical interpretations”, we consider only the latter type of “interpretations”. A parametrical interpretation I of a first order language L in a consistent theory T′ (formulated in another first order language L′) consists of the following formulas:(i) a unary formula C(p) (i.e. a formula with one designated free variable p), which is used to denote the range of parameters,(ii) a binary formula U(p, x), which is intended to denote the pth universe for each parameter p,(iii) an (n + 1)-ary formula Fp(p, x1 …, xn) for each n-ary predicate symbol P in L,such that the formulas (∃p)C(p) and (∀p)(C(p)→(∃x)U(p, x)) are provable in T". Then, given a formula A in L and a parameter p, we define the interpretation Ip (A ) of A by I at p to be the formula which is obtained from A by replacing every atomic subformula P(*, …, *) in A by Fp(p, *,…,*), and relativizing every occurrence of quantifiers in A by U(p, * ). A sentence A in L is said to be I-provable in T′ if the sentence (∀p) (C(p)→ Ip(A)) is provable in T′. Then, it is obvious that every provable sentence in L is I-provable in T′. This is a basic result of “parametrical interpretations” and is used to prove the “consistency” of a theory T in L by showing that every axiom of T is I-provable in T′ when I is said to be a parametrical interpretation of T in T′. As is shown above, the word “interpretation” is used in the following three senses: interpretations of languages, interpretations of formulas and interpretations of theories. So, in this introduction we let the word “interpretation” denote “interpretation of languages”, for short.

The incompleteness theorems

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Mathematical Reasoning ◽  
Order Theory ◽  
Point Of View ◽  
Incompleteness Theorem ◽  
Bertrand Russell ◽  
First Order ◽  
First Order Theory ◽  
Incompleteness Theorems ◽  
Special Case ◽  
Gödel’S Incompleteness Theorems

In this chapter we prove that the structure N = (ℕ|+, · , 1) has a first-order theory that is undecidable. This is a special case of Gödel’s First Incompleteness theorem. This theorem implies that any theory (not necessarily first-order) that describes elementary arithmetic on the natural numbers is necessarily undecidable. So there is no algorithm to determine whether or not a given sentence is true in the structure N. As we shall show, the existence of such an algorithm leads to a contradiction. Gödel’s Second Incompleteness theorem states that any decidable theory (not necessarily first-order) that can express elementary arithmetic cannot prove its own consistency. We shall make this idea precise and discuss the Second Incompleteness theorem in Section 8.5. Gödel’s First Incompleteness theorem is proved in Section 8.3. Although they are purely mathematical results, Gödel’s Incompleteness theorems have had undeniable philosophical implications. Gödel’s theorems dispelled commonly held misconceptions regarding the nature of mathematics. A century ago, some of the most prominent mathematicians and logicians viewed mathematics as a branch of logic instead of the other way around. It was thought that mathematics could be completely formalized. It was believed that mathematical reasoning could, at least in principle, be mechanized. Alfred North Whitehead and Bertrand Russell envisioned a single system that could be used to derive and enumerate all mathematical truths. In their three-volume Principia Mathematica, Russell and Whitehead rigorously define a system and use it to derive numerous known statements of mathematics. Gödel’s theorems imply that any such system is doomed to be incomplete. If the system is consistent (which cannot be proved within the system by Gödel’s Second theorem), then there necessarily exist true statements formulated within the system that the system cannot prove (by Gödel’s First theorem). This explains why the name “incompleteness” is attributed to these theorems and why the title of Gödel’s 1931 paper translates (from the original German) to “On Formally Undecidable Propositions of Principia Mathematica and Related Systems” (translated versions appear in both [13] and [14]). Depending on one’s point of view, it may or may not be surprising that there is no algorithm to determine whether or not a given sentence is true in N.

More undecidable lattices of Steinitz exchange systems

2002 ◽  
Vol 67 (2) ◽  
pp. 859-878 ◽  
Author(s):  
L. R. Galminas ◽  
John W. Rosenthal
Keyword(s):  
Number Theory ◽  
Order Theory ◽  
Closed Field ◽  
Order Number ◽  
Infinite Dimensional ◽  
First Order ◽  
First Order Theory ◽  
Finite Dimensional ◽  
Order Arithmetic ◽  
Countably Infinite

AbstractWe show that the first order theory of the lattice <ω(S) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System S has logical complexity at least that of first order number theory and that the first order theory of the lattice (S∞) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System S∞ has logical complexity exactly that of first order number theory. Thus, for example, the lattice of finite dimensional subspaces of a standard copy of ⊕ωQ interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice (V∞) of c.e. subspaces of a fully effective ℵ0-dimensional vector space V∞ and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field F∞ of countably infinite transcendence degree each have logical complexity that of first order number theory.

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