Logical Compactness and Constraint Satisfaction Problems
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Ultrafilter Extensions and the Standard Translation
Notes from Lecture 4: Ultrafilter Extensions and the Standard Translation Eric Pacuit∗ March 1, 2012 1 Ultrafilter Extensions Definition 1.1 (Ultrafilter) Let W be a non-empty set. An ultrafilter on W is a set u ⊆ }(W ) satisfying the following properties: 1. ; 62 u 2. If X; Y 2 u then X \ Y 2 u 3. If X 2 u and X ⊆ Y then Y 2 u. 4. For all X ⊆ W , either X 2 u or X 2 u (where X is the complement of X in W ) / A collection u0 ⊆ }(W ) has the finite intersection property provided for each X; Y 2 u0, X \ Y 6= ;. Theorem 1.2 (Ultrafilter Theorem) If a set u0 ⊆ }(W ) has the finite in- tersection property, then u0 can be extended to an ultrafilter over W (i.e., there is an ultrafilter u over W such that u0 ⊆ u. Proof. Suppose that u0 has the finite intersection property. Then, consider the set u1 = fZ j there are finitely many sets X1;:::Xk such that Z = X1 \···\ Xkg: That is, u1 is the set of finite intersections of sets from u0. Note that u0 ⊆ u1, since u0 has the finite intersection property, we have ; 62 u1, and by definition u1 is closed under finite intersections. Now, define u2 as follows: 0 u = fY j there is a Z 2 u1 such that Z ⊆ Y g ∗UMD, Philosophy. Webpage: ai.stanford.edu/∼epacuit, Email: [email protected] 1 0 0 We claim that u is a consistent filter: Y1;Y2 2 u then there is a Z1 2 u1 such that Z1 ⊆ Y1 and Z2 2 u1 such that Z2 ⊆ Y2. -
An Introduction to Nonstandard Analysis 11
AN INTRODUCTION TO NONSTANDARD ANALYSIS ISAAC DAVIS Abstract. In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demon- strate how theorems in standard analysis \transfer over" to nonstandard anal- ysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis. 1. Introduction For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount. Archimedes derived the formula for the area of a circle by thinking of a circle as a polygon with infinitely many infinitesimal sides [1]. In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change. G.W. Leibniz's derivation of calculus made extensive use of “infinitesimal” numbers, which were both nonzero but small enough to add to any real number without changing it noticeably. Although intuitively clear, infinitesi- mals were ultimately rejected as mathematically unsound, and were replaced with the common -δ method of computing limits and derivatives. However, in 1960 Abraham Robinson developed nonstandard analysis, in which the reals are rigor- ously extended to include infinitesimal numbers and infinite numbers; this new extended field is called the field of hyperreal numbers. The goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis. In this paper, we will explore the construction and various uses of nonstandard analysis. In section 2 we will introduce the notion of an ultrafilter, which will allow us to do a typical ultrapower construction of the hyperreal numbers. -
The Nonstandard Theory of Topological Vector Spaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 THE NONSTANDARDTHEORY OF TOPOLOGICAL VECTOR SPACES BY C. WARD HENSON AND L. C. MOORE, JR. ABSTRACT. In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems ; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which tesults. Introduction. Let Ml be a set theoretical structure and let *JR be an enlarge- ment of M. Let (E, 0) be a topological vector space in M. §§1 and 2 of this paper are devoted to the elementary nonstandard theory of (F, 0). In particular, in §1 the concept of 0-finiteness for elements of *E is introduced and the nonstandard hull of (E, 0) (relative to *3R) is defined. §2 introduces the concept of 0-bounded- ness for elements of *E. In §5 the elementary nonstandard theory of locally convex spaces is developed by investigating the mapping in *JK which corresponds to a given pairing. In §§6 and 7 we make use of this theory by providing nonstandard treatments of two aspects of the existing standard theory. In §6, Luxemburg's characterization of the pre-nearstandard elements of *E for a normed space (E, p) is extended to Hausdorff locally convex spaces (E, 8). This characterization is used to prove the theorem of Grothendieck which gives a criterion for the completeness of a Hausdorff locally convex space. -
1. Introduction in a Topological Space, the Closure Is Characterized by the Limits of the Ultrafilters
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 08–37 THE ULTRAFILTER CLOSURE IN ZF GONC¸ALO GUTIERRES Abstract: It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with uX (A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈U]}. However, it is possible to built a topological space X for which uX 6= kX , but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {R}. Keywords: Ultrafilter Theorem, Ultrafilter Closure. AMS Subject Classification (2000): 03E25, 54A20. 1. Introduction In a topological space, the closure is characterized by the limits of the ultrafilters. Although, in the absence of the Axiom of Choice, this is not a fact anymore. -
First Order Logic and Nonstandard Analysis
First Order Logic and Nonstandard Analysis Julian Hartman September 4, 2010 Abstract This paper is intended as an exploration of nonstandard analysis, and the rigorous use of infinitesimals and infinite elements to explore properties of the real numbers. I first define and explore first order logic, and model theory. Then, I prove the compact- ness theorem, and use this to form a nonstandard structure of the real numbers. Using this nonstandard structure, it it easy to to various proofs without the use of limits that would otherwise require their use. Contents 1 Introduction 2 2 An Introduction to First Order Logic 2 2.1 Propositional Logic . 2 2.2 Logical Symbols . 2 2.3 Predicates, Constants and Functions . 2 2.4 Well-Formed Formulas . 3 3 Models 3 3.1 Structure . 3 3.2 Truth . 4 3.2.1 Satisfaction . 5 4 The Compactness Theorem 6 4.1 Soundness and Completeness . 6 5 Nonstandard Analysis 7 5.1 Making a Nonstandard Structure . 7 5.2 Applications of a Nonstandard Structure . 9 6 Sources 10 1 1 Introduction The founders of modern calculus had a less than perfect understanding of the nuts and bolts of what made it work. Both Newton and Leibniz used the notion of infinitesimal, without a rigorous understanding of what they were. Infinitely small real numbers that were still not zero was a hard thing for mathematicians to accept, and with the rigorous development of limits by the likes of Cauchy and Weierstrass, the discussion of infinitesimals subsided. Now, using first order logic for nonstandard analysis, it is possible to create a model of the real numbers that has the same properties as the traditional conception of the real numbers, but also has rigorously defined infinite and infinitesimal elements. -
An Introduction to Nonstandard Analysis
An Introduction to Nonstandard Analysis Jeffrey Zhang Directed Reading Program Fall 2018 December 5, 2018 Motivation • Developing/Understanding Differential and Integral Calculus using infinitely large and small numbers • Provide easier and more intuitive proofs of results in analysis Filters Definition Let I be a nonempty set. A filter on I is a nonempty collection F ⊆ P(I ) of subsets of I such that: • If A; B 2 F , then A \ B 2 F . • If A 2 F and A ⊆ B ⊆ I , then B 2 F . F is proper if ; 2= F . Definition An ultrafilter is a proper filter such that for any A ⊆ I , either A 2 F or Ac 2 F . F i = fA ⊆ I : i 2 Ag is called the principal ultrafilter generated by i. Filters Theorem Any infinite set has a nonprincipal ultrafilter on it. Pf: Zorn's Lemma/Axiom of Choice. The Hyperreals Let RN be the set of all real sequences on N, and let F be a fixed nonprincipal ultrafilter on N. Define an (equivalence) relation on RN as follows: hrni ≡ hsni iff fn 2 N : rn = sng 2 F . One can check that this is indeed an equivalence relation. We denote the equivalence class of a sequence r 2 RN under ≡ by [r]. Then ∗ R = f[r]: r 2 RNg: Also, we define [r] + [s] = [hrn + sni] [r] ∗ [s] = [hrn ∗ sni] The Hyperreals We say [r] = [s] iff fn 2 N : rn = sng 2 F . < is defined similarly. ∗ ∗ A subset A of R can be enlarged to a subset A of R, where ∗ [r] 2 A () fn 2 N : rn 2 Ag 2 F : ∗ ∗ ∗ Likewise, a function f : R ! R can be extended to f : R ! R, where ∗ f ([r]) := [hf (r1); f (r2); :::i] The Hyperreals A hyperreal b is called: • limited iff jbj < n for some n 2 N. -
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness Theorem for First Order Logic There are many proofs of the Completeness Theorem for First Order Logic. We follow here a version of Henkin's proof, as presented in the Handbook of Mathe- matical Logic. It contains a method for reducing certain problems of first-order logic back to problems about propositional logic. We give independent proof of Compactness Theorem for propositional logic. The Compactness Theorem for first-order logic and L¨owenheim-Skolem Theorems and the G¨odelCompleteness Theorem fall out of the Henkin method. 1.1 Compactness Theorem for Propositional Logic Let L = L(P; F; C) be a first order language with equality. We assume that the sets P, F, C are infinitely enumerable. We define a propositional logic within it as follows. Prime formulas We consider a subset P of the set F of all formulas of L. Intuitively these are formulas of L which are not direct propositional com- bination of simpler formulas, that is, atomic formulas (AF) and formulas beginning with quantifiers. Formally, we have that P = fA 2 F : A 2 AF or A = 8xB; A = 9xB for B 2 Fg: Example 1.1 The following are primitive formulas. R(t1; t2); 8x(A(x) ):A(x)); (c = c); 9x(Q(x; y) \ 8yA(y)). The following are not primitive formulas. (R(t1; t2) ) (c = c)); (R(t1; t2) [ 8x(A(x) ):A(x)). Given a set P of primitive formulas we define in a standard way the set P F of propositional formulas as follows. -
Foundations of Mathematics
Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions . -
SELECTIVE ULTRAFILTERS and Ω −→ (Ω) 1. Introduction Ramsey's
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 3067{3071 S 0002-9939(99)04835-2 Article electronically published on April 23, 1999 SELECTIVE ULTRAFILTERS AND ! ( ! ) ! −→ TODD EISWORTH (Communicated by Andreas R. Blass) Abstract. Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59{ 111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in L(R)is -Ramsey with respect to every selective ultrafilter . In this paper, we showU that the large cardinal assumption cannot be weakened.U 1. Introduction Ramsey's theorem [5] states that for any n !,iftheset[!]n of n-element sets of natural numbers is partitioned into finitely many∈ pieces, then there is an infinite set H ! such that [H]n is contained in a single piece of the partition. The set H is said⊆ to be homogeneous for the partition. We can attempt to generalize Ramsey's theorem to [!]!, the collection of infinite subsets of the natural numbers. Let ! ( ! ) ! abbreviate the statement \for every [!]! there is an infinite H ! such−→ that either [H]! or [H]! = ." UsingX⊆ the axiom of choice, it is easy⊆ to partition [!]! into two⊆X pieces in such∩X a way∅ that any two infinite subsets of ! differing by a single element lie in different pieces of the partition. Obviously such a partition can admit no infinite homogeneous set. Thus under the axiom of choice, ! ( ! ) ! is false. Having been stymied in this naive−→ attempt at generalization, we have several obvious ways to proceed. A natural project would be to attempt to find classes of partitions of [!]! that do admit infinite homogeneous sets. -
Shelah's Singular Compactness Theorem 0
SHELAH’S SINGULAR COMPACTNESS THEOREM PAUL C. EKLOF Abstract. We present Shelah’s famous theorem in a version for modules, together with a self-contained proof and some ex- amples. This exposition is based on lectures given at CRM in October 2006. 0. Introduction The Singular Compactness Theorem is about an abstract notion of “free”. The general form of the theorem is as follows: If λ is a singular cardinal and M is a λ-generated module such that enough < κ-generated submodules are “free” for sufficiently many regular κ < λ, then M is “free”. Of course, for this to have any chance to be a theorem (of ZFC) there need to be assumptions on the notion of “free”. These are detailed in the next section along with a precise statement of the theorem (Theorem 1.4), including a precise definition of “enough”. Another version of the Singular Compactness Theorem—with a dif- ferent definition of “enough”—is given at the start of section 3. The proof of Theorem 1.4 is given in sections 3 and 4; some examples and applications are given in sections 2 and 5. First a few words about the history of the theorem. 0.1 History. In 1973 Saharon Shelah proved that the Whitehead problem for abelian groups of cardinality ℵ1 is undecidable in ZFC; in particular he showed under the assumption of the Axiom of Con- structibility, V = L, that all Whitehead groups of cardinality ℵ1 are free (see [13]). His argument easily extended, by induction, to prove that for all n ∈ ω, Whitehead groups of cardinality ℵn are free. -
Adequate Ultrafilters of Special Boolean Algebras 347
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 174, December 1972 ADEQUATEULTRAFILTERS OF SPECIAL BOOLEANALGEBRAS BY S. NEGREPONTIS(l) ABSTRACT. In his paper Good ideals in fields of sets Keisler proved, with the aid of the generalized continuum hypothesis, the existence of countably in- complete, /jf^-good ultrafilters on the field of all subsets of a set of (infinite) cardinality ß. Subsequently, Kunen has proved the existence of such ultra- filters, without any special set theoretic assumptions, by making use of the existence of certain families of large oscillation. In the present paper we succeed in carrying over the original arguments of Keisler to certain fields of sets associated with the homogeneous-universal (and more generally with the special) Boolean algebras. More specifically, we prove the existence of countably incomplete, ogood ultrafilters on certain powers of the ohomogeneous-universal Boolean algebras of cardinality cl and on the a-completions of the ohomogeneous-universal Boolean algebras of cardinality a, where a= rc-^ > w. We then develop a method that allows us to deal with the special Boolean algebras of cardinality ct= 2""". Thus, we prove the existence of an ultrafilter p (which will be called adequate) on certain powers S * of the special Boolean algebra S of cardinality a, and the ex- istence of a specializing chain fC«: ß < a\ for oa, such that C snp is /3+- good and countably incomplete for ß < a. The corresponding result on the existence of adequate ultrafilters on certain completions of the special Boolean algebras is more technical. These results do not use any part of the generalized continuum hypothesis. -
Ultraproducts (Handout April, 2011) G
Ultraproducts (Handout April, 2011) G. Eric Moorhouse, University of Wyoming Let Mξ ξ∈X be an indexed collection of structures over a common language L. The { } product Π = Qξ∈X Mξ is also a structure over this same language L. Example. Suppose each Mξ is a field. We take for L the language of rings (which serves also as the common language of fields). Thus L has constants 0 and 1, and two binary operations: + and . (Possibly other symbols as well: a unary function × symbol for additive inverses, and a unary function for multiplicative inverses, per- − haps represented by the superscript symbol −1. Let’s not worry about these for now.) Then Π is a structure over the same language L: addition and multiplication are de- fined componentwise. We interpret the symbol 0 in Π as the element having 0 in each component (or more correctly, in component ξ we take the zero element of Mξ, as the interpretation of the symbol 0 may vary from component to component. Similarly 1 is interpreted in Π as the element whose ξ-component is interpreted as the identity element of Mξ. If each Mξ is a field, then it is easily checked that Π = Qξ∈X Mξ is a ring; but not a field for X > 1, since Π has zero divisors. To correct this we need | | ultrafilters. Let U be an ultrafilter on X. Then M = Π/U is also a structure over L; this is known as an ultraproduct of the Mξ’s. Elements of M = Π/U are equivalence classes of Π where two elements x = (xξ )ξ∈X and y = (yξ )ξ∈X in Π are equivalent if ‘almost all’ coordinates agree (relative to U).