
Matthias Aschenbrenner
et al.
Published online:
19 October 2017
Published in print:
06 June 2017
Online ISBN:
9781400885411
Print ISBN:
9780691175423
Contents
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
B.1 Structures and their Definable Sets B.1 Structures and their Definable Sets
-
Model-theoretic structures Model-theoretic structures
-
Conventions and notations Conventions and notations
-
Notes and comments Notes and comments
-
-
B.2 Languages B.2 Languages
-
Languages Languages
-
L-structures L-structures
-
Products Products
-
Notes and comments Notes and comments
-
-
B.3 Variables and Terms B.3 Variables and Terms
-
Variables Variables
-
Terms Terms
-
Variable-free terms Variable-free terms
-
Names Names
-
Substitution Substitution
-
Generators Generators
-
Unique readability Unique readability
-
-
B.4 Formulas B.4 Formulas
-
Formulas Formulas
-
Substitution Substitution
-
Truth and definability Truth and definability
-
Formulas of a special form Formulas of a special form
-
Maps preserving formulas Maps preserving formulas
-
More on substitution More on substitution
-
Notes and comments Notes and comments
-
-
B.5 Elementary Equivalence and Elementary Substructures B.5 Elementary Equivalence and Elementary Substructures
-
Elementary equivalence Elementary equivalence
-
Elementary substructures Elementary substructures
-
-
PROOF. After enlarging A to a parameter set in N of size k, we may assume that jAj = k. For every B ⊆ N and s, let {B;s be the set of all LB-formulas ’(x) with x a quantifiable variable of sort s such that N j= 9x’. For every ’ 2 {B;s choose a b’ 2 Ns with N j= ’(b’s)0, and let B:= fb’ : ’ 2 {B;sg. For each b 2 Bs and quantifiable variable x sof 0sort s the LB-formula x = b is in {B;s, so b {= bx=b 2 B, hence Bs ⊆ B. Setting ss00B0 := (B), we have jBj 6 jB0j 6 max jBj; jLj. We now inductively define an increasing sequence A0 ⊆ A1 ⊆ … of parameter sets inM Sby A0 := A and An+1 := A0n, and put M := n An. Then jAnj = k for each n, hence also jMj = k. By Proposition B.5.8, M is the underlying set of an elementary substructure of N. ∈Λ PROOF. After enlarging A to a parameter set in N of size k, we may assume that jAj = k. For every B ⊆ N and s, let {B;s be the set of all LB-formulas ’(x) with x a quantifiable variable of sort s such that N j= 9x’. For every ’ 2 {B;s choose a b’ 2 Ns with N j= ’(b’s)0, and let B:= fb’ : ’ 2 {B;sg. For each b 2 Bs and quantifiable variable x sof 0sort s the LB-formula x = b is in {B;s, so b {= bx=b 2 B, hence Bs ⊆ B. Setting ss00B0 := (B), we have jBj 6 jB0j 6 max jBj; jLj. We now inductively define an increasing sequence A0 ⊆ A1 ⊆ … of parameter sets inM Sby A0 := A and An+1 := A0n, and put M := n An. Then jAnj = k for each n, hence also jMj = k. By Proposition B.5.8, M is the underlying set of an elementary substructure of N. ∈Λ
-
Direct unions Direct unions
-
Algebraic closure and definable closure Algebraic closure and definable closure
-
-
B.6 Models and the Compactness Theorem B.6 Models and the Compactness Theorem
-
Models Models
-
Logical consequence Logical consequence
-
-
PROOF. If Fl [fn1 6= 0 : n > 1g j= σ, then compactness yields N 2 N such that Fl [fn1 6= 0 : n = 1; : : : ;Ng j= σ, so σ holds in all .fields of characteristic p > N∈Λ PROOF. If Fl [fn1 6= 0 : n > 1g j= σ, then compactness yields N 2 N such that Fl [fn1 6= 0 : n = 1; : : : ;Ng j= σ, so σ holds in all .fields of characteristic p > N∈Λ
-
Completeness and compactness Completeness and compactness
-
Notes and comments Notes and comments
-
-
B.7 Ultraproducts and Proof of the Compactness Theorem B.7 Ultraproducts and Proof of the Compactness Theorem
-
COROLLARY B.7.8. If U is an ultrafilter on *, then the diagonal embedding ofN into N*=U is elementary. COROLLARY B.7.8. If U is an ultrafilter on *, then the diagonal embedding ofN into N*=U is elementary.
-
Proof of the Compactness Theorem Proof of the Compactness Theorem
-
Functoriality of reduced products Functoriality of reduced products
-
Notes and comments Notes and comments
-
-
B.8 Some Uses of Compactness B.8 Some Uses of Compactness
-
Diagrams Diagrams
-
Substructures and universal sentences Substructures and universal sentences
-
The Löwenheim-Skolem Theorem The Löwenheim-Skolem Theorem
-
-
B.9 Types and Saturated Structures B.9 Types and Saturated Structures
-
Separating types Separating types
-
Types over a parameter set Types over a parameter set
-
Saturated structures Saturated structures
-
Notes and comments Notes and comments
-
-
B.10 Model Completeness B.10 Model Completeness
-
Existentially closed models Existentially closed models
-
Model companions and model completions Model companions and model completions
-
-
commute. Using the definitions and Corollary B.8.4 one easily shows: commute. Using the definitions and Corollary B.8.4 one easily shows:
-
F := (b; c) 2 M1 ×M2 : nb = f1(a); nc = –f2(a) for some n > 1 and a 2 A; F := (b; c) 2 M1 ×M2 : nb = f1(a); nc = –f2(a) for some n > 1 and a 2 A;
-
Notes and comments Notes and comments
-
-
B.11 Quantifier Elimination B.11 Quantifier Elimination
-
A general elimination theorem A general elimination theorem
-
Application to QE Application to QE
-
A test for quantifier elimination A test for quantifier elimination
-
QE tests using saturation QE tests using saturation
-
QE and definable closure QE and definable closure
-
-
PROOF. Replacing Z by Z [(Mx n π(Z))×Mywe arrange that π(Z) = Mx. Take a quantifier-free formula ’(x; y) that defines Z inM. Set PROOF. Replacing Z by Z [(Mx n π(Z))×Mywe arrange that π(Z) = Mx. Take a quantifier-free formula ’(x; y) that defines Z inM. Set
-
Notes and comments Notes and comments
-
-
B.12 Application to Algebraically Closed and Real Closed Fields B.12 Application to Algebraically Closed and Real Closed Fields
-
Algebraically closed fields Algebraically closed fields
-
Real closed fields Real closed fields
-
Notes and comments Notes and comments
-
-
B.13 Structures Without the Independence Property B.13 Structures Without the Independence Property
-
Independence dimension Independence dimension
-
VC duality VC duality
-
Notes and comments Notes and comments
-
-
-
-
-
-
End Matter
Appendix B Basic Model Theory
Get access
Pages
724–786
-
Published:June 2017
Cite
Aschenbrenner, Matthias, Lou van den Dries, and Joris van der Hoeven, ' Basic Model Theory', Asymptotic Differential Algebra and Model Theory of Transseries (Princeton, NJ , 2017; online edn, Princeton Scholarship Online, 19 Oct. 2017), https://doi.org/10.23943/princeton/9780691175423.005.0002, accessed 19 Apr. 2025.
You do not currently have access to this chapter.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMetrics
View Metrics
Metrics
Total Views
4
3
Pageviews
1
PDF Downloads
Since 4/1/2023
Month: | Total Views: |
---|---|
April 2023 | 3 |
August 2024 | 1 |
Citations
Altmetrics
More from Oxford Academic
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.