LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 9
|
|
- Clare Dickerson
- 5 years ago
- Views:
Transcription
1 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 9 6 Lecture These memoirs begin with material presented during lecture 5, but omitted from the memoir of that lecture. We observed that the notion of a strict linear order can be characterized by a first-order sentence, the conjunction of the following conditions: ( x) x < x (irreflexivity) ( x)( y)( z)(x < y (y < z x < z)) (transitivity) ( x)( y)(x y (x < y y < x)) (comparability) We noted that for every natural number n there is a unique, up to isomorphism, linear order on the set [n] = {1,..., n}. On the other hand, Exercise 1 Show that there are 2 ℵ0 set {1, 2, 3,...}. linear orders, up to isomorphism, on the 1 We gave several examples of infinite linear orders: 1. N, < (the natural numbers with their usual order); 2. Z, < (the integers with their usual order); 3. Q, < (the rational numbers with their usual order); 4. R, < (the real numbers with their usual order); 5. N,, where i j if and only if i is even and j is odd, or the parity of i and j is the same and i < j, (the natural numbers with an unusual order). 2 We examined first-order conditions that distinguish among these orderings. 1. ( x)( y) y < x (there is a least element) 2. ( x)( y) x < y (there is a greatest element) 3. ( x)(( w)x < w ( y)(x < y ( z) (x < z z < y))) (discrete) 4. ( x)( y)(x < y ( z)(x < z z < y)) (dense) 5. ( x)(( y)y < x ( z)(z < x ( w)(z < w w < x))) (there is a limit point) Observe that orders (1)1 and 2 are discrete and have no limit point, moreover, the first of these has a least element while the second does not. The orders (1)3, 4, and 5 all have at least one limit point, moreover, the first two are dense while the last is discrete. Exercise 2 Show that Q, < R, <.
2 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 10 Here begins the memoir of lecture 6. Thus far we have been exploring the expressive power of first-order logic by looking at classes of structures which are first-order definable. We say a class of structures K is an elementary class if and only if there is a first-sentence ϕ such that K = Mod(ϕ) and we say K is and extended elementary class if and only if there is a set of first-order sentences Σ such that K = Mod(Σ). Another approach to the study of expressive power is via consideration of the relations which are first-definable on a fixed structure. If ϕ(x 1,..., x n ) is a formula with free variables among x 1,..., x n, and A is a structure which interprets all the non-logical symbols occurring in ϕ, then ϕ[a] denotes the n-ary relation defined by ϕ on A, that is, ϕ[a] = {< a 1,..., a n > A = ϕ[a 1,..., a n ]}. We presented solutions to problems and in Enderton, which deal with definable collections of structures and definability within a fixed structure respectively. Let A = A, P A be a structure for a language with a binary relation P and with no further relation, function, or constant symbols other than identity. We will often use A, rather than A, to denote the universe of A when no confusion is likely to result. If f is a function with domain A and range contained in A, that is, a function from A into A, we say that P A is the graph of f if and only if for all a, b A, a, b P A f(a) = b. Let α be the sentence x yp xy x y z((p xy P xz) y = z). Note that Mod(α) is the collection of all structures A such that P A is the graph of a function from A into A. Let β be the sentence x y z((p xz P yz) x = y). Note that Mod(α β) is the collection of all structures A such that P A is the graph of an injection (that is, 1-1 function) from A into A. Let γ be the sentence x yp yx. Note that Mod(α γ) is the collection of all structures A such that P A is the graph of a surjection from A onto A. Finally, note that Mod(α β γ) is the collection of all structures A such that P A is the graph of a permutation of A, that is, a bijection from A onto A. We next considered definability within the fixed structure N = N, +, where N = {0, 1, 2,...} and + and are the usual arithmetic operations on N. We considered the definability of simple sets and relations on N per exercise (a) y(x + y = y)[n] = {0}.
3 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 11 (b) y(x y = y)[n] = {1}. (c) z( w(z w = w) x + z = y)[n] = { m, n n = m + 1}. (d) z( w(z + w w) x + z = y)[n] = { m, n m < n}. We next considered the structures N = N, < and Z = Z, < where Z = {... 1, 0, 1,...}. We noted that y (y < x)[n] = {0} and asked whether {0} is definable in Z. The general sentiment was negative, but we agreed that we d need a new idea to settle the question. To this end, we introduced the notion of an isomorphism of one structure onto another and of an automorphism, that is, an isomorphism of a structure onto itself. We say that f is an isomorphism of A onto B if and only if f is a bijection of A onto B and for all a, b A a, b P A f(a), f(b) P B. An automorphism of A is an isomorphism of A onto A. We write Aut(A) for the set of automorphisms of A. We stated the following theorem, which says that first-order logic satisfies a natural desideratum for a language to be logical it does not distinguish between structurally identical models. Theorem 1 (Isomorphism Theorem) Suppose f is an isomorphism of A onto B. Then for every formula α(x 1,..., x n ), with at most the variables indicated free, and for all a 1,..., a n A, A = α[(x 1 a 1 ),..., (x n a n )] B = α[(x 1 f(a 1 )),..., (x n f(a n ))]. As a corollary to the Isomorphism Theorem, we have the Corollary 1 (Automorphism Theorem) If f is an automorphism of A and α(x) is a formula with at most x free, then for all a A, a α[a] f(a) α[a]. We applied the automorphism theorem to show that the only sets definable in Z = Z, < are Z and. This follows from the observation that for every p Z the function f p Aut(Z), where f p (q) = p + q, for all q Z. We continued to go over exercises from Enderton, Section 2.2. Exercise provided another opportunity to apply the Automorphism Theorem to establish an undefinability result. We showed that the addition relation, that is, { p, q, r p + q = r}, is not definable in the structure N = N,. We first observed that the ordering relation m < n is definable in terms of addition by the formula z(z + z z x + z = y). Therefore, it suffices to show that the ordering relation is not definable in N = N,. For this, it suffices to exhibit an automorphism h of N = N, which is not order preserving, that is, for some p, q we have p < q but h(q) h(p). We showed that there is an automorphism h of N = N, such that h(2) = 3 and h(3) = 2, thereby completing the exercise. We observed that any permutation of the prime numbers can be extended to an automorphism of N = N,. We concluded that there are uncountably many automorphisms of N = N,.
4 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 12 Exercise 3 1. Let B = {f f is a bijection from N onto N} and let C = {f f : N {0, 1}}. Show that there is a bijection of B onto C. 2. With C as above show that there is a bijection from C onto Aut( Q, < ). 7 Lecture We continued our study of the expressive power of first-order logic. Today we focused on finite structures. We recalled our earlier observation that for every finite directed graph A, there is a sentence ψ such that for all graphs directed graphs B, B = ψ A = B. Exercise 4 Let A be a finite structure that interprets an infinite collection of relation symbols R n where R n is n-ary. Show that for every structure B, B = Th(A) A = B. With Exercise we meet another measure of expressive power that is specially suited to measure expressiveness in the context of finite structures. Given a first-order sentence α, we define the spectrum of α as follows: Spec(α) = {n N A(card(A) = n and A = α}. We are asked to exhibit a sentence α whose spectrum is the set of positive even numbers. Here is such a sentence: x Rxx x y(rxy Ryx) x y z(rxz z = y). The sentence is true in a structure just in case that structure is a loop-free undirected graph which is 1-regular, that is, all vertices have degree 1. It is easy to see that such a graph must have an even number of elements, and that for every even number n, there is such a graph of size n. We proceeded to give an example of a sentence ϕ whose spectrum is the set of perfect squares. The sentence used one ternary relation symbol R and one unary relation symbol F. A structure A satisfies ϕ if and only if R A is the graph of a bijection of F A F A onto A. We may chose ϕ to be the conjunction of the following sentences. ( x)( y)( z)( w)((f x F y) (Rxyw w = z)) ( x)( y)( v)( w)( z)((rxyz Rvwz) (F x F y x = v y = w)) ( z)( x)( y)rxyz
5 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 13 Exercise 5 Let L be the first-order order language with only one ternary relation symbol F and two unary predicate symbols P and Q and identity. Give an example of a sentence γ of L whose spectrum is the set of powers of 2. We briefly discussed the Spectrum Problem: Is the collection of firstorder spectra closed under complementation? We remarked that this problem is equivalent to the closure of NEXP under complementation, a deep question in the theory of computational complexity. Recall that a set X N is cofinite if and only if N X is finite. We began to show that for every first-order sentence α in the language of directed graphs Spec(α) is cofinite or Spec( α) is cofinite. This result is a corollary of the following two central theorems about first-order logic. Theorem 2 (Compactness Theorem) If a set of first-order sentences is finitely satisfiable, then it is satisfiable. Theorem 3 (Downward Löwenheim-Skolem Theorem) If a countable set of first-order sentences has an infinite model, it has a countable model.
Final Exam (100 points)
Final Exam (100 points) Honor Code: Each question is worth 10 points. There is one bonus question worth 5 points. In contrast to the homework assignments, you may not collaborate on this final exam. You
More informationVAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0. Contents
VAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0 BENJAMIN LEDEAUX Abstract. This expository paper introduces model theory with a focus on countable models of complete theories. Vaught
More informationMore Model Theory Notes
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A M with A M finite, and any
More informationMarch 3, The large and small in model theory: What are the amalgamation spectra of. infinitary classes? John T. Baldwin
large and large and March 3, 2015 Characterizing cardinals by L ω1,ω large and L ω1,ω satisfies downward Lowenheim Skolem to ℵ 0 for sentences. It does not satisfy upward Lowenheim Skolem. Definition sentence
More information2.2 Lowenheim-Skolem-Tarski theorems
Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore
More informationIntroduction to Model Theory
Introduction to Model Theory Charles Steinhorn, Vassar College Katrin Tent, University of Münster CIRM, January 8, 2018 The three lectures Introduction to basic model theory Focus on Definability More
More informationIntroduction to Model Theory
Introduction to Model Theory Jouko Väänänen 1,2 1 Department of Mathematics and Statistics, University of Helsinki 2 Institute for Logic, Language and Computation, University of Amsterdam Beijing, June
More informationMODEL THEORY FOR ALGEBRAIC GEOMETRY
MODEL THEORY FOR ALGEBRAIC GEOMETRY VICTOR ZHANG Abstract. We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert s Nullstellensatz, Noether- Ostrowski, and Hilbert s
More informationMathematical Logic (IX)
Mathematical Logic (IX) Yijia Chen 1. The Löwenheim-Skolem Theorem and the Compactness Theorem Using the term-interpretation, it is routine to verify: Theorem 1.1 (Löwenheim-Skolem). Let Φ L S be at most
More informationHerbrand Theorem, Equality, and Compactness
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order
More informationAMS regional meeting Bloomington, IN April 1, 2017
Joint work with: W. Boney, S. Friedman, C. Laskowski, M. Koerwien, S. Shelah, I. Souldatos University of Illinois at Chicago AMS regional meeting Bloomington, IN April 1, 2017 Cantor s Middle Attic Uncountable
More informationLöwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)
Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1
More informationMATH 3300 Test 1. Name: Student Id:
Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your
More informationQualifying Exam Logic August 2005
Instructions: Qualifying Exam Logic August 2005 If you signed up for Computability Theory, do two E and two C problems. If you signed up for Model Theory, do two E and two M problems. If you signed up
More informationMeta-logic derivation rules
Meta-logic derivation rules Hans Halvorson February 19, 2013 Recall that the goal of this course is to learn how to prove things about (as opposed to by means of ) classical first-order logic. So, we will
More informationAbstract model theory for extensions of modal logic
Abstract model theory for extensions of modal logic Balder ten Cate Stanford, May 13, 2008 Largely based on joint work with Johan van Benthem and Jouko Väänänen Balder ten Cate Abstract model theory for
More informationSpecial Topics on Applied Mathematical Logic
Special Topics on Applied Mathematical Logic Spring 2012 Lecture 04 Jie-Hong Roland Jiang National Taiwan University March 20, 2012 Outline First-Order Logic Truth and Models (Semantics) Logical Implication
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationPreliminaries. Introduction to EF-games. Inexpressivity results for first-order logic. Normal forms for first-order logic
Introduction to EF-games Inexpressivity results for first-order logic Normal forms for first-order logic Algorithms and complexity for specific classes of structures General complexity bounds Preliminaries
More informationFundamentals of Model Theory
Fundamentals of Model Theory William Weiss and Cherie D Mello Department of Mathematics University of Toronto c 2015 W.Weiss and C. D Mello 1 Introduction Model Theory is the part of mathematics which
More informationBasics of Model Theory
Chapter udf Basics of Model Theory bas.1 Reducts and Expansions mod:bas:red: defn:reduct mod:bas:red: prop:reduct Often it is useful or necessary to compare languages which have symbols in common, as well
More informationTeddy Einstein Math 4320
Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationDiscrete Mathematics
Discrete Mathematics Yi Li Software School Fudan University June 12, 2017 Yi Li (Fudan University) Discrete Mathematics June 12, 2017 1 / 16 Review Soundness and Completeness Theorem Compactness Theorem
More informationModel theory of algebraically closed fields
U.U.D.M. Project Report 2017:37 Model theory of algebraically closed fields Olle Torstensson Examensarbete i matematik, 15 hp Handledare: Vera Koponen Examinator: Jörgen Östensson Oktober 2017 Department
More informationWhat is the right type-space? Humboldt University. July 5, John T. Baldwin. Which Stone Space? July 5, Tameness.
Goals The fundamental notion of a Stone space is delicate for infinitary logic. I will describe several possibilities. There will be a quiz. Infinitary Logic and Omitting Types Key Insight (Chang, Lopez-Escobar)
More information.. Discrete Mathematics. Yi Li. June 9, Software School Fudan University. Yi Li (Fudan University) Discrete Mathematics June 9, / 15
Discrete Mathematics Yi Li Software School Fudan University June 9, 2013 Yi Li (Fudan University) Discrete Mathematics June 9, 2013 1 / 15 Review Soundness and Completeness Theorem Compactness Theorem
More informationAlgebras with finite descriptions
Algebras with finite descriptions André Nies The University of Auckland July 19, 2005 Part 1: FA-presentability A countable structure in a finite signature is finite-automaton presentable (or automatic)
More informationMASTERS EXAMINATION IN MATHEMATICS
MASTERS EXAMINATION IN MATHEMATICS PURE MATH OPTION, Spring 018 Full points can be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth 0 points.
More informationFINITE MODEL THEORY (MATH 285D, UCLA, WINTER 2017) LECTURE NOTES IN PROGRESS
FINITE MODEL THEORY (MATH 285D, UCLA, WINTER 2017) LECTURE NOTES IN PROGRESS ARTEM CHERNIKOV 1. Intro Motivated by connections with computational complexity (mostly a part of computer scientice today).
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationOctober 12, Complexity and Absoluteness in L ω1,ω. John T. Baldwin. Measuring complexity. Complexity of. concepts. to first order.
October 12, 2010 Sacks Dicta... the central notions of model theory are absolute absoluteness, unlike cardinality, is a logical concept. That is why model theory does not founder on that rock of undecidability,
More informationExpressiveness of predicate logic: Some motivation
Expressiveness of predicate logic: Some motivation In computer science the analysis of the expressiveness of predicate logic (a.k.a. first-order logic) is of particular importance, for instance In database
More informationPart II Logic and Set Theory
Part II Logic and Set Theory Theorems Based on lectures by I. B. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationChapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationCOMPLETENESS OF THE RANDOM GRAPH: TWO PROOFS
COMPLETENESS OF THE RANDOM GRAPH: TWO PROOFS EUGENIA FUCHS Abstract. We take a countably infinite random graph, state its axioms as a theory in first-order logic, and prove its completeness in two distinct
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationNotes on ordinals and cardinals
Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}
More informationA function is a special kind of relation. More precisely... A function f from A to B is a relation on A B such that. f (x) = y
Functions A function is a special kind of relation. More precisely... A function f from A to B is a relation on A B such that for all x A, there is exactly one y B s.t. (x, y) f. The set A is called the
More informationScott Sentences in Uncountable Structures
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 14 Scott Sentences in Uncountable Structures Brian Tyrrell Trinity College Dublin Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj
More informationPRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY
Iranian Journal of Fuzzy Systems Vol. 10, No. 3, (2013) pp. 103-113 103 PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY S. M. BAGHERI AND M. MONIRI Abstract. We present some model theoretic results for
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More informationPropositional and Predicate Logic - XIII
Propositional and Predicate Logic - XIII Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - XIII WS 2016/2017 1 / 22 Undecidability Introduction Recursive
More informationMASTERS EXAMINATION IN MATHEMATICS
MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION FALL 2010 Full points can be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth 20 points.
More informationMorley s Proof. Winnipeg June 3, 2007
Modern Model Theory Begins Theorem (Morley 1965) If a countable first order theory is categorical in one uncountable cardinal it is categorical in all uncountable cardinals. Outline 1 2 3 SELF-CONSCIOUS
More informationDefinition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.
4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be
More informationSolutions to Unique Readability Homework Set 30 August 2011
s to Unique Readability Homework Set 30 August 2011 In the problems below L is a signature and X is a set of variables. Problem 0. Define a function λ from the set of finite nonempty sequences of elements
More informationCOMP 409: Logic Homework 5
COMP 409: Logic Homework 5 Note: The pages below refer to the text from the book by Enderton. 1. Exercises 1-6 on p. 78. 1. Translate into this language the English sentences listed below. If the English
More informationDisjoint n-amalgamation
October 13, 2015 Varieties of background theme: the role of infinitary logic Goals 1 study n- toward 1 existence/ of atomic models in uncountable cardinals. 2 0-1-laws 2 History, aec, and Neo-stability
More informationAxioms of separation
Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More information0-1 Laws for Fragments of SOL
0-1 Laws for Fragments of SOL Haggai Eran Iddo Bentov Project in Logical Methods in Combinatorics course Winter 2010 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and
More informationA Vaught s conjecture toolbox
Chris Laskowski University of Maryland 2 nd Vaught s conjecture conference UC-Berkeley 1 June, 2015 Everything begins with the work of Robert Vaught. Everything begins with the work of Robert Vaught. Fix
More informationCHAPTER 2. FIRST ORDER LOGIC
CHAPTER 2. FIRST ORDER LOGIC 1. Introduction First order logic is a much richer system than sentential logic. Its interpretations include the usual structures of mathematics, and its sentences enable us
More informationMath 557: Mathematical Logic. Homework #7 (Revised) SOLUTIONS
Math 557: Mathematical Logic Homework #7 (Revised) October 11, 2000 SOLUTIONS October 30, 2000 1. Let L be a finite language with identity. Let M be a finite normal L-structure. Construct an L-sentence
More informationNON-ISOMORPHISM INVARIANT BOREL QUANTIFIERS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 NON-ISOMORPHISM INVARIANT BOREL QUANTIFIERS FREDRIK ENGSTRÖM AND PHILIPP SCHLICHT Abstract. Every
More informationDiscrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland
Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationIncomplete version for students of easllc2012 only. 6.6 The Model Existence Game 99
98 First-Order Logic 6.6 The Model Existence Game In this section we learn a new game associated with trying to construct a model for a sentence or a set of sentences. This is of fundamental importance
More informationFriendly Logics, Fall 2015, Lecture Notes 5
Friendly Logics, Fall 2015, Lecture Notes 5 Val Tannen 1 FO definability In these lecture notes we restrict attention to relational vocabularies i.e., vocabularies consisting only of relation symbols (or
More informationRepresenting Scott Sets in Algebraic Settings
Representing Scott Sets in Algebraic Settings Alf Dolich Kingsborough Community College Julia F. Knight University of Notre Dame Karen Lange Wellesley College David Marker University of Illinois at Chicago
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationPosets, homomorphisms and homogeneity
Posets, homomorphisms and homogeneity Peter J. Cameron and D. Lockett School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract Jarik Nešetřil suggested
More informationA Hanf number for saturation and omission: the superstable case
A Hanf number for saturation and omission: the superstable case John T. Baldwin University of Illinois at Chicago Saharon Shelah The Hebrew University of Jerusalem Rutgers University April 29, 2013 Abstract
More informationGeneral Notation. Exercises and Problems
Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.
More informationMath 225A Model Theory. Speirs, Martin
Math 5A Model Theory Speirs, Martin Autumn 013 General Information These notes are based on a course in Metamathematics taught by Professor Thomas Scanlon at UC Berkeley in the Autumn of 013. The course
More informationMATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. II - Model Theory - H. Jerome Keisler
ATHEATCS: CONCEPTS, AND FOUNDATONS Vol. - odel Theory - H. Jerome Keisler ODEL THEORY H. Jerome Keisler Department of athematics, University of Wisconsin, adison Wisconsin U.S.A. Keywords: adapted probability
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationCORES OF COUNTABLY CATEGORICAL STRUCTURES
Logical Methods in Computer Science Vol. 3 (1:2) 2007, pp. 1 16 www.lmcs-online.org Submitted Sep. 23, 2005 Published Jan. 25, 2007 CORES OF COUNTABLY CATEGORICAL STRUCTURES MANUEL BODIRSKY Institut für
More informationThe Absoluteness of Constructibility
Lecture: The Absoluteness of Constructibility We would like to show that L is a model of V = L, or, more precisely, that L is an interpretation of ZF + V = L in ZF. We have already verified that σ L holds
More informationProperties of Relational Logic
Computational Logic Lecture 8 Properties of Relational Logic Michael Genesereth Autumn 2011 Programme Expressiveness What we can say in First-Order Logic And what we cannot Semidecidability and Decidability
More informationPrinciples of Real Analysis I Fall I. The Real Number System
21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous
More information1 Model theory; introduction and overview
1 Model theory; introduction and overview Model theory, sometimes described as algebra with quantifiers is, roughly, mathematics done with attention to definability. The word can refer to definability
More informationRepresenting Scott Sets in Algebraic Settings
Wellesley College Wellesley College Digital Scholarship and Archive Faculty Research and Scholarship 8-2015 Representing Scott Sets in Algebraic Settings Alf Dolich Julia F. Knight Karen Lange klange2@wellesley.edu
More informationContinuum Harvard. April 11, Constructing Borel Models in the. Continuum Harvard. John T. Baldwin. University of Illinois at Chicago
April 11, 2013 Today s Topics 1 2 3 4 5 6 Pseudo-minimal 7 Further Applications Section 1: { Models in L ω1,ω L ω1,ω satisfies downward Löwenheim Skolem to ℵ 0 for sentences. It does not satisfy upward
More informationThe rise and fall of uncountable models
Chris Laskowski University of Maryland 2 nd Vaught s conjecture conference UC-Berkeley 2 June, 2015 Recall: Every counterexample Φ to Vaught s Conjecture is scattered. Recall: Every counterexample Φ to
More informationFoundations of Mathematics Worksheet 2
Foundations of Mathematics Worksheet 2 L. Pedro Poitevin June 24, 2007 1. What are the atomic truth assignments on {a 1,..., a n } that satisfy: (a) The proposition p = ((a 1 a 2 ) (a 2 a 3 ) (a n 1 a
More informationJune 28, 2007, Warsaw
University of Illinois at Chicago June 28, 2007, Warsaw Topics 1 2 3 4 5 6 Two Goals Tell Explore the notion of Class as a way of examining extensions of first order logic Ask Does the work on generalized
More informationLECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel
LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is
More informationAlgebraic structures I
MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More informationarxiv: v1 [cs.pl] 19 May 2016
arxiv:1605.05858v1 [cs.pl] 19 May 2016 Domain Theory: An Introduction Robert Cartwright Rice University Rebecca Parsons ThoughtWorks, Inc. Moez AbdelGawad SRTA-City Hunan University This monograph is an
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More informationA Framework for Inductive Logic 1
A Framework for Inductive Logic 1 Eric Martin University of New South Wales Daniel Osherson Princeton University May 1, 2003 Page 1 of 70 1 This work was supported by Australian Research Council Grant
More informationCharacterizing First Order Logic
Characterizing First Order Logic Jared Holshouser, Originally by Lindstrom September 16, 2014 We are following the presentation of Chang and Keisler. 1 A Brief Review of First Order Logic Definition 1.
More informationA1 Logic (25 points) Using resolution or another proof technique of your stated choice, establish each of the following.
A1 Logic (25 points) Using resolution or another proof technique of your stated choice, establish each of the following. = ( x)( y)p (x, y, f(x, y)) ( x)( y)( z)p (x, y, z)). b. (6.25 points) Γ = ( x)p
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationSection 0. Sets and Relations
0. Sets and Relations 1 Section 0. Sets and Relations NOTE. Mathematics is the study of ideas, not of numbers!!! The idea from modern algebra which is the focus of most of this class is that of a group
More informationThe Vaught Conjecture Do uncountable models count?
The Vaught Conjecture Do uncountable models count? John T. Baldwin Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago May 22, 2005 1 Is the Vaught Conjecture model
More informationMathematical Logic II. Dag Normann The University of Oslo Department of Mathematics P.O. Box Blindern 0316 Oslo Norway
Mathematical Logic II Dag Normann The University of Oslo Department of Mathematics P.O. Box 1053 - Blindern 0316 Oslo Norway December 21, 2005 Contents 1 Classical Model Theory 6 1.1 Embeddings and isomorphisms...................
More informationFoundations of Mathematics
Foundations of Mathematics L. Brian Lawrence Department of Mathematics George Mason University Fairfax, VA 22030 4444 U.S.A. e mail: blawrenc@mail.gmu.edu January 1, 2007 Preface This set of notes is an
More informationCategoricity Without Equality
Categoricity Without Equality H. Jerome Keisler and Arnold W. Miller Abstract We study categoricity in power for reduced models of first order logic without equality. 1 Introduction The object of this
More informationDiscrete dynamics on the real line
Chapter 2 Discrete dynamics on the real line We consider the discrete time dynamical system x n+1 = f(x n ) for a continuous map f : R R. Definitions The forward orbit of x 0 is: O + (x 0 ) = {x 0, f(x
More informationA generalization of modal definability
A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models
More informationProf. Ila Varma HW 8 Solutions MATH 109. A B, h(i) := g(i n) if i > n. h : Z + f((i + 1)/2) if i is odd, g(i/2) if i is even.
1. Show that if A and B are countable, then A B is also countable. Hence, prove by contradiction, that if X is uncountable and a subset A is countable, then X A is uncountable. Solution: Suppose A and
More informationIntroduction to Model theory Zoé Chatzidakis CNRS (Paris 7) Notes for Luminy, November 2001
Introduction to Model theory Zoé Chatzidakis CNRS (Paris 7) Notes for Luminy, November 2001 These notes aim at giving the basic definitions and results from model theory. My intention in writing them,
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #10: Sequences and Summations Based on materials developed by Dr. Adam Lee Today s Topics Sequences
More informationLINDSTRÖM S THEOREM SALMAN SIDDIQI
LINDSTRÖM S THEOREM SALMAN SIDDIQI Abstract. This paper attempts to serve as an introduction to abstract model theory. We introduce the notion of abstract logics, explore first-order logic as an instance
More information