DOCSLIB.ORG

A Logician's Toolbox

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Logician's Toolbox A Logician’s Toolbox 461: An Introduction to Mathematical Logic Spring 2009 We recast/introduce notions which arise everywhere in mathematics. All proofs are left as exercises. 0 Notations from set theory •{ 1, 2} = {2, 1} = {1, 2, 1} • The ordered pair (a, b) is the set {a, {a, b}}. One has (a, b) = (c, d) iff a = c and b = d. • The Cartesian product of two sets A and B is A × B = {(a, b): a ∈ A, b ∈ B} • The power set of A is the set of all subsets of A, P (A) = {X : X ⊆ A} • When A and B are sets, BA denotes the set of all functions A → B. • 0 ∈ N! 1 Functions Definition 1.1 (function). Function A function from A to B is a relation f on A × B such that: ∀a ∈ A ∀b, b0 ∈ B, (a, b) ∈ f ∧ (a, b0) ∈ f → b = b0 We then write f : A → B, and b = f(a) for (a, b) ∈ f. When talking about a function, it is always very important to make clear its domain dom f and its codomain cod f (A and B respectively in the definition above). To indicate the mapping, the following notation is customary: f : A → B x 7→ f(x) 1 Definition 1.2 (composition). Composition Let f : A → B, g : B → C be functions. Their composition is the function g ◦ f : A → C defined by: (a, c) ∈ g ◦ f if ∃b ∈ B (a, b) ∈ f ∧ (b, c) ∈ g In other words, c = (g ◦ f)(a) iff there is b ∈ B such that b = f(a) and c = g(b) iff c = g(f(a)). Properties 1.3. Composition is associative. If f : A → B, g : B → C, h : C → D are functions, then h ◦ (g ◦ f) = (h ◦ g) ◦ f. 1.1 Injections and Surjections Definition 1.4 (injection). Injection A function f : A → B is injective/an injection if ∀a, a0 ∈ A, f(a) = f(a0) → a = a0 Notation 1.5. Injectivity of f is sometimes denoted f : A,→ B. Properties 1.6. Let f : A → B, g : B → C be functions. • If f and g are injective, so is g ◦ f. • If g ◦ f is injective, so is f. Counter-example 1.7. Let f be the inclusion map from {0} into {0, 1}, and g be the constant map from {0, 1} to {0}. Then g ◦ f is the identity function of {0}, clearly injective, but g isn’t. Definition 1.8 (surjection). Surjection A function f : A → B is surjective/a surjection if ∀b ∈ B ∃a ∈ A, b = f(a) Notation 1.9. Surjectivity of f is sometimes denoted f : A B. Properties 1.10. Let f : A → B, g : B → C be functions. • If f and g are surjective, so is g ◦ f. • If g ◦ f is surjective, so is g. Counter-example 1.11. In counter-example 1.7, g◦f is surjective, but f isn’t. 1.2 Bijections Definition 1.12 (bijection). Bijection A function f : A → B is bijective/a bijection if it is both injective and surjective. Properties 1.13. Let f : A → B, g : B → C be functions. • If f and g are bijective, so is g ◦ f. • If g ◦ f is bijective, then f is injective and g is surjective. Proposition 1.14. Let f : A → B be a bijection. Then there exists a unique function g : B → A such that g ◦ f = IdA and f ◦ g = IdB. Moreover, g is a bijection; it is called the reciprocal bijection of f and denoted f −1. 2 1.3 Image and Pre-image Sets Definition 1.15 (Image set). Image set Let f : A → B be a function and X ⊆ A be a subset of A. The image of X under f is f(X) = {f(x): x ∈ A} Properties 1.16. Let f : A → B be a function and X, Y ⊆ A subsets of A. • f(X ∩ Y ) ⊆ f(X) ∩ f(Y ) • f(X ∪ Y ) = f(X) ∪ f(Y ) Counter-example 1.17. Let f : {0, 1} → {0} be the constant function; set X = {0} and Y = {1}. Then X ∩ Y = ∅ but f(X) ∩ f(Y ) = {0} 6= ∅. Exercise 1.18. Let f : A → B be a function and g defined as follows: g : P (A) → P (B) X 7→ f(X) 1. Show that f is injective iff g is. 2. Show that f is surjective iff g is. Definition 1.19 (Pre-image set). Pre-image Let f : A → B be a function and X ⊆ B be a subset of B. The pre-image of X under f is f −1(X) = {x ∈ A : f(x) ∈ X} Remark 1.20. f −1 as a function from B to A makes sense only when f is a bijection; f −1 as a function from P (B) to P (A) is always defined. It is the case that when f is bijective and X ⊆ B, one has f −1(X) = f −1(X) | {z } | {z } Image under f −1 Pre−image under f Properties 1.21. Let f : A → B be a function and X, Y ⊆ B subsets of B. • f −1(X ∩ Y ) = f −1(X) ∩ f −1(Y ) • f −1(X ∪ Y ) = f −1(X) ∪ f −1(Y ) The following is dual to Exercise 1.18. Exercise 1.22. Let f : A → B be a function and h defined as follows: h : P (B) → P (A) X 7→ f −1(X) 1. Show that f is injective iff h is surjective. 2. Show that f is surjective iff h is injective. 3 1.4 Their interplay Exercise 1.23. Let f : A → B be a function. 1. Let X ⊆ A. Show that f −1(f(X)) ⊆ X. 2. Suppose that f is injective. Show ∀X ⊆ A, f −1(f(X)) = X 3. Suppose ∀X ⊆ A, f −1(f(X)) = X Show that f is injective. Exercise 1.24. Let f : A → B be a function. 1. Let X ⊆ B. Show that X ⊆ f(f −1(X). 2. Suppose that f is surjective. Show ∀X ⊆ B, f(f −1(X)) = X. 3. Suppose ∀X ⊆ B, f(f −1(X)) = X. Show that f is surjective. 2 Equivalence Relations 2.1 Equivalence Relations Definition 2.1 (equivalence relation). Equivalence relation An equivalence re- lation on a set A is a relation ∼⊆ A × A which is reflexive, symmetric, and transitive. We write of course a ∼ b instead of (a, b) ∈∼. Hence the definition is: •∀ a ∈ A, a ∼ a (reflexivity) •∀ a, b ∈ A, a ∼ b → b ∼ a (symmetry) •∀ a, b, c ∈ A, a ∼ b ∧ b ∼ c → a ∼ c (transitivity) Definition 2.2 (equivalence class). Let ∼ be an equivalence relation on a set A. Let a ∈ A. Then the equivalence class of a modulo ∼ is the set [a]∼ = {b ∈ A : a ∼ b} One may drop the subscript ∼ if there is no risk of ambiguity. Properties 2.3. Let ∼ be an equivalence relation on a set A. Let a, b ∈ A. Then a ∼ b iff a ∈ [b]∼ iff b ∈ [a]∼ iff [a]∼ ∩ [b]∼ 6= ∅. Definition 2.4 (quotient set). Quotient set Let ∼ be an equivalence relation on a set A. The quotient set of A by ∼ is the set of all equivalence classes: A/ ∼= {[a]∼ : a ∈ A} 4 2.2 Partitions Definition 2.5 (partition). Partition Let A be a set. A partition of A is a subset Π of P (A) such that: • ∅ 6∈ Π (no member of Π is empty) • S Π = A (members of Π cover A) •∀ X, Y ∈ Π,X ∩Y 6= ∅ → X = Y (distinct members of Π don’t overlap). The typical example is that of the set of classes of an equivalence relation. Remark 2.6. Let ∼ be an equivalence relation on A. Then A/ ∼ is a partition of A. This is no coincidence. Proposition 2.7. Let A be a set. Then there is a bijection between the set of equivalence relations on A and the set of partitions of A. 3 Order Relations 3.1 Pre-orderings Definition 3.1 (pre-ordering). Pre-ordering A pre-ordering on a set A is a relation - which is reflexive and transitive. •∀ a ∈ A, a - a (reflexivity) •∀ a, b, c ∈ A, a - b ∧ b - c → a - c (transitivity) These tend to occur in a mathematician’s everyday life, though they are fairly loose relations. Notice that no one said that elements a and b could always be compared: perhaps neither a - b nor b - a holds. The dullest example of a pre-ordering is the empty relation, that is no two elements are in relation. When a - b ∧ b - a, one says that a and b can be compared. 3.2 Partial Orderings Definition 3.2 (ordering, poset). OrderingPoset An ordering on a set A is a pre-ordering which is anti-symmetric. •∀ a ∈ A, a - a (reflexivity) •∀ a, b ∈ A, a b ∧ b a → a = b (anti-symmetry) •∀ a, b, c ∈ A, a - b ∧ b - c → a - c (transitivity) One then says that (A, ) is a poset. 5 Of course there is a way to retrieve an ordering from any pre-ordering; it suffices to say that elements which both lie above each other must actually be equal. This is done by “factoring out" the natural equivalence relation. Proposition 3.3. Let - be a pre-ordering on a set A. Then • The relation ∼ defined by a ∼ b if a - b ∧ b - a is an equivalence relation on A. • The relation defined by [a]∼ [b]∼ if a - b is an ordering relation on A/ ∼. • The relation defined above is the only order relation on A/ ∼ which is compatible with -.
Load more

Recommended publications
  • Sets and Classes As Many
    SETS AND CLASSES AS MANY by John L. Bell INTRODUCTION Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated in the diagram: Individuals Sets Classes If on the other hand we insist—as we shall here—that classes are to be taken in the sense of multitudes, pluralities, or classes as many, then no class can be an individual and so, in particular, the concept of set will need to be redefined. Here by “class as many” we have in mind what Erik Stenius refers to in [5] as set of, which he defines as follows: If we start from a Universe of Discourse given in advance, then we may define a set-of things as being many things in this UoD or just one thing - or even no things, if we want to introduce this way of speaking. Stenius draws a sharp distinction between this concept and that of 2 set as a thing. As he says, The distinction between a set-as-a-thing and a set of corresponds to the Russellian distinction between a “class as one” and a “class as many” (Principles of Mathematics, p. 76). Only I use the expressions ‘set-as-a- thing” and “set-of” instead, in order to stress the (grammatical and) ontological character of the distinction; and also because of the difficulty that a set-of need not consist of many things; it can comprise just one thing or no things.
    [Show full text]
  • Preliminary Knowledge on Set Theory
    Huan Long Shanghai Jiao Tong University Basic set theory Relation Function Spring 2018 Georg Cantor(1845-1918) oGerman mathematician o Founder of set theory Bertrand Russell(1872-1970) oBritish philosopher, logician, mathematician, historian, and social critic. Ernst Zermelo(1871-1953) oGerman mathematician, foundations of mathematics and hence on philosophy David Hilbert (1862-1943) o German mathematicia, one of the most influential and universal mathematicians of the 19th and early 20th centuries. Kurt Gödel(1906-1978) oAustrian American logician, mathematician, and philosopher. ZFC not ¬CH . Paul Cohen(1934-2007)⊢ oAmerican mathematician, 1963: ZFC not CH,AC . Spring 2018 ⊢ Spring 2018 By Georg Cantor in 1870s: A set is an unordered collection of objects. ◦ The objects are called the elements, or members, of the set. A set is said to contain its elements. Notation: ◦ Meaning that: is an element of the set A, or, Set A contains ∈ . Important: ◦ Duplicates do not matter. ◦ Order does not matter. Spring 2018 a∈A a is an element of the set A. a A a is NOT an element of the set A. Set of sets {{a,b},{1, 5.2}, k} ∉ the empty set, or the null set, is set that has no elements. A B subset relation. Each element of A is also an element of B. ∅ A=B equal relation. A B and B A. ⊆ A B ⊆ ⊆ A B strict subset relation. If A B and A B ≠ |A| cardinality of a set, or the number of distinct elements. ⊂ ⊆ ≠ Venn Diagram V U A B U Spring 2018 { , , , } a {{a}} ∈ ∉ {{ }} ∅ ∉∅ {3,4,5}={5,4,3,4} ∅ ∈ ∅ ∈ ∅ S { } ∅⊆ S S ∅ ⊂ ∅ |{3, 3, 4, {2, 3},{1,2,{f}} }|=4 ⊆ Spring 2018 Union Intersection Difference Complement Symmetric difference Power set Spring 2018 Definition Let A and B be sets.
    [Show full text]
  • Elements of Set Theory
    Elements of Set Theory P´eterMekis Department of Logic, ELTE Budapest December 11, 2017 Preface This is an as yet incomplete draft of what is going to be the lecture notes for a course under the same title for students of the Logic and the Theory of Science MA program of ELTE Budapest. The text is written for students that are logically and philosophically minded, but do not necessarily have a mathematical background. However, it requires familiarity with the basics of first-order logic. A reader who is skilled in reading mathematical texts will find most of the proofs too detailed. Note that any proof can be skipped and taken as an exercise to avoid boredom. Instead of the more well-known system of Zermelo{Fraenkel set theory, these notes use the framework of G¨odel{Bernays set theory (GB). Beyond the personal taste of the author, the reason for this choice is that there is a follow-up of this course under the title Alternative set theories, the basic task of which is to investigate the philosophical motivations and implications of various versions of the set/class distinction, which is at the heart of any axiomatic set theory, and which is more convenient to discuss in a language where one can quantify over classes. A mathematically minded reader may be surprised by the length of discussing certain basic topics, even without any apparent practical profit, and even at the cost of giving a rather sketchy presentation of more advanced results. An example is the detailed com- parison of three different ways to represent natural numbers in GB.
    [Show full text]
  • 2.5. INFINITE SETS Now That We Have Covered the Basics of Elementary
    2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after Georg Cantor laid out the core principles of his new theory of sets in the late 19th century, his work led him to a trove of controversial and groundbreaking results related to the cardinalities of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous theorem on power sets and his famous diagonal argument, both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of mathematics – the Continuum Hypothesis, which posits that the cardinality of the continuum (i.e. the set of all points on a line) is equal to that of the power set of the set of natural numbers. Lastly, we conclude this section with a foray into transfinite arithmetic, an extension of the usual arithmetic with finite numbers that includes operations with so-called aleph numbers – the cardinal numbers of infinite sets. If all of this sounds rather outlandish at the moment, don’t be surprised. The properties of infinite sets can be highly counter-intuitive and you may likely be in total disbelief after encountering some of Cantor’s theorems for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit interval (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague Richard Dedekind: “I see it, but I don’t believe it!” The Tricky Nature of Infinity Throughout the ages, human beings have always wondered about infinity and the notion of uncountability.
    [Show full text]
  • Logicism, Interpretability, and Knowledge of Arithmetic
    ZU064-05-FPR RSL-logicism-final 1 December 2013 21:42 The Review of Symbolic Logic Volume 0, Number 0, Month 2013 Logicism, Interpretability, and Knowledge of Arithmetic Sean Walsh Department of Logic and Philosophy of Science, University of California, Irvine Abstract. A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure- based notions of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. Received February 2013 Acknowledgements: This is material coming out of my dissertation, and I would thus like to thank my advisors, Dr. Michael Detlefsen and Dr.
    [Show full text]
  • Set Theory Handout
    Salem State College MAT 420 Set Theory L. Pedro Poitevin October 25, 2007 1 Foundations De¯nition (Informal). We de¯ne a set A to be any unordered collection of objects, e.g., f3; 8; 5; 1g is a set. If x is an object, we say that x is an element of A or x 2 A if x lies in the collection; otherwise we say that x is not an element of A and write x2 = A. For instance, 3 2 f1; 2; 3; 4; 5g but 7 2= f1; 2; 3; 4; 5g. Axiom 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is an element of B. De¯nition (Equality of sets). Two sets A and B are equal|in notation A = B|if and only if every element of A is an element of B and vice versa. Axiom 2 (Empty set). There exists a set ;, known as the empty set, which contains no elements, i.e. for every object x we have x2 = ;. Exercise. Prove that there is only one empty set. Lemma 1 (Single choice). Let A be a nonempty set. Then there exists an object x such that x 2 A. Axiom 3 (Singletons). If a is an object, then there exists a set fag whose only element is a. Axiom 4 (Pairwise union). Given any two sets A and B, there exists a set A [ B called the union of A and B, whose elements are all objects which are elements of A or B.
    [Show full text]
  • Chapter Five: Infinities
    CHAPTER FIVE: INFINITIES Two sets A and B are said to be equinumerous, written A ≈ B, iff there is a bijection from A to B. It follows that a set is finite iff it is equinumerous with a natural number. As we've observed (Chapter 3, problem 1), equinumerosity is an equiv- alence relation on the powerset of any set. It is not hard to show that for any set A, }(A) ≈ 2A: the bijection in question maps each subset of A to its characteristic function (with respect to A). Intuitively speaking, equinumerosity may seem to amount to \having the same number of members". As we soon will see, this intuition is essentially on the mark in the case of finite sets. But when the sets involved are infinite, intuition may fail us. For example, all the following sets can be shown to be (pairwise) equinumerous: !, ! × !, the set Z of integers, and the set Q of rational numbers.1 Not all infinite sets are equinumerous! To put it imprecisely but sug- gestively, there are \different sizes of infinity". For example, as Cantor famously proved, ! 6≈ I, where I is the set of real numbers from 0 to 1. It is beyond the scope of this book to consider how the real numbers are modelled set-theoretically, but for our purposes it will suffice to think of I as the set of \decimal expansions", i.e. the set of functions from ! to 10 (= f0; 1; 2; 3; 4; 5; 6; 7; 8; 9g) excluding the ones which for some natural number n assign 9 to every natural number greater than or equal to n.2 The proof is surprisingly simple: suppose f is an injection from ! to I.
    [Show full text]
  • Chapter 6: Cardinal Numbers and the Axiom of Choice Equinumerosity March 18 & 20, 2014
    Elements of Set Theory Chapter 6: Cardinal Numbers and The Axiom of Choice Equinumerosity March 18 & 20, 2014 Lecturer: Fan Yang 1/28 2/28 What is the size of a set? X = f , , , g Y = f , , , , g ! = f0; 1; 2; 3; 4;::: g Do A and B have the same size? Does A have more elements than B? Are there exactly as many houses as people? Y has more elements than X. Yes, since there are 5 houses and 5 people. The infinite set ! has more elements than the finite sets X and Y . Yes, since there is a one-to-one correspondence between the two Do ! and Z have the same size? What about Q and Z? sets. 0 1 2 3 ! Definition 6.1 -3 -2 -1 0 1 2 3 Z A set A is said to be equinumerous or equipotent to a set B (written 3 1 11 − 2 2 4 A ≈ B) iff there is a bijection from A onto B. -3 -2 -1 0 1 2 3 Q A bijection from A onto B is also called a one-to-one correspondence Given two infinite sets A and B, how to compare their sizes? between sets A and B. 3/28 4/28 Example 6.1: Let A = fa; b; c; dg and B = fa; b; cg. Then A 6≈ B, since there is no bijection from A onto B. In general, for any two finite sets X and Y , if Y ⊂ X, then X 6≈ Y . Consider the following infinite sets. Clearly, Even ⊂ !.
    [Show full text]
  • Transfinite Numbers What Is Infinity?
    GENERAL I ARTICLE Transfinite Numbers What is Infinity? S M Srivastava In a series of revolutionary articles written during the last quarter of the nineteenth century, the great Ger­ man mathematician Georg Cantor removed the age-old mistrust of infinity and created an exceptionally beau­ tiful and useful theory of transfinite numbers. This is an introductory article on this topic.1 Introduction S M Srivastava is with the Indian Statistical How many points are there on a line? Which is more in number­ Institute, Calcutta. He points on a line or lines in a plane? These are some natural received his PhD from the questions that have occurred to us sometime or the other. It is Indian Statistical Institute interesting to note the difference between the two questions. Do in 1980. His research we have to know how many points and lines there are to answer interests are in Descriptive Set Theory. the second question? Even primitive man could see whether he had more cows than his neighbour without knowing the actual number of cows they had-he would just pair off his cows with The author would like to his neighbour's. So, to compare the size of two sets, we can start gratefully acknowledge the pairing off their elements. If one of them exhausts first we say help received from 5 C Bagchi. that that particular set has fewer elements than the other one. T Krishnan. B V Rao and H Otherwise we say that the two sets have the same number of el­ Sarbadhikari while preparing ements.
    [Show full text]
  • Solutions: Equinumerosity
    Peter Smith, Introduction to G¨odel'sTheorems, 2nd edition 2013 (CUP) Solutions: Equinumerosity Notation: ∆ ⊆ Γ says that ∆ is a subset of Γ in the sense that every member of ∆ is a member of Γ, which allows ∆ = Γ. We use ∆ ⊂ Γ when ∆ is a proper subset of Γ, so ∆ ⊆ Γ ^ ∆ 6= Γ 1. Two sets ∆ and Γ are said to be equinumerous iff there is a one-one correspondence between them, i.e. there is some bijection f : ∆ ! Γ. For a simple reality check, show that equinu- merosity is an equivalence relation. In other words, writing ‘∆ ≈ Γ’ for ‘∆ is equinumerous to Γ’, show that (a)∆ ≈ ∆. (b) If ∆ ≈ Γ then Γ ≈ ∆. (c) If ∆ ≈ Γ and Γ ≈ Θ, then ∆ ≈ Θ. (a) The identity function that maps an element in ∆ to the very same element in ∆ is a bijection! (b) If there is a bijection f : ∆ ! Γ, then it has an inverse f −1 :Γ ! ∆ which is also a bijection. (c) We just need to recall that if f : ∆ ! Γ and g :Γ ! Θ are bijections, so is their composition g ◦ f : ∆ ! Θ. [See the exercise set on Functions.] 2. Show: (a) A finite set cannot be equinumerous with one of its proper subsets (i.e. with some some subset strictly contained in it). [Hint: argue for the contrapositive, i.e. if equinumerous, then not finite.] (b) An infinite set can be equinumerous with one of its proper subsets. (c) The set of natural numbers is equinumerous with the set of ordered pairs of natural numbers. (d) The set of natural numbers is equinumerous with the set of positive rational numbers.
    [Show full text]
  • MATH 320 SET THEORY Contents 0. Prelude 0.1. Some Historical
    MATH 320 SET THEORY BURAK KAYA Abstract. These are the lecture notes I used for a 14-week introductory set theory class I taught at the Department of Mathematics of Middle East Tech- nical University during Spring 2018. In order to determine the course content and prepare the lecture notes, I mainly used the textbook by Hrbacek and Jech [1], which I also listed as a supplementary resource for the course. Contents 0. Prelude 0.1. Some historical remarks. 0.2. The language of set theory and well-formed formulas 0.3. What are sets anyway? 0.4. Classes vs. Sets 0.5. Notational remarks 1. Some axioms of ZFC and their elementary consequences 1.1. And G said, \Let there be sets"; and there were sets. 1.2. Constructing more sets 2. From Pairs to Products 2.1. Relations 2.2. Functions 2.3. Products and sequences 2.4. To choose or not to choose 3. Equivalence Relations and Order Relations 3.1. Equivalence relations, partitions and transversals. 3.2. A Game of Thrones, Prisoners and Hats. 3.3. Order relations 3.4. Well-orders. 3.5. Well-founded relations and the Axiom of Foundation 4. Natural Numbers 4.1. The construction of the set of natural numbers 4.2. Arithmetic on the set of natural numbers 5. Equinumerosity 5.1. Finite sets 5.2. To infinity and beyond 6. Construction of various number systems 6.1. Integers 6.2. Rational numbers 6.3. Real numbers 7. Ordinal numbers 7.1. How do the ordinals look like? BURAK KAYA 7.2.
    [Show full text]
  • Arithmetic, Abstraction, and the Frege Quantifier
    ARITHMETIC,ABSTRACTION, AND THE FREGE QUANTIFIER G. Aldo Antonelli University of California, Davis 2011 ASL Meeting Berkeley, CA ALDO ANTONELLI, UC DAVIS ARITHMETIC WITH THE FREGE QUANTIFIER DATE: 2011 ASL MEETING SLIDE: 1/27 LOGICISM AND ABSTRACTION Goal of the talk: to present a formalization of first-order arithmetic characterized by the following: 1 Natural numbers are identified with abstracta of the equinumerosity relation; 2 Abstraction itself receives a deflationary construal — abstracts have no special ontological status. 3 Logicism is articulated in a non-reductionist fashion: rather than reducing arithmetic to principles whose logical character is questionable, we take seriously Frege’s idea that cardinality is a logical notion. 4 The formalization uses two main technical tools: A first-order (binary) cardinality quantifier F expressing “For every A there is a (distinct) B’s”; An abstraction operator Num assigning first-level objects to predicates. 5 The logicist banner is then carried by the quantifier, rather than by Hume’s Principle. 6 Finally, the primary target of the formalization are the cardinal properties of the natural numbers, rather than the structural ones. ALDO ANTONELLI, UC DAVIS ARITHMETIC WITH THE FREGE QUANTIFIER DATE: 2011 ASL MEETING SLIDE: 2/27 ARITHMETICAL REDUCTION STRATEGIES ... The main formalization strategies for first-order arithmetic: The Peano-Dedekind approach: numbers are primitive, their properties given by the usual axioms; The Frege-Russell tradition: natural numbers are identified with equinumerosity classes; The Zermelo-von Neumann implementation: natural numbers are identified with particular representatives of those equivalence classes, e.g. ;; f;g; ff;gg; fff;ggg;::: or ;; f;g; f;; f;gg; f;; f;g; f;; f;ggg;::: Note: Numbers are not always members of the equivalence classes they represent — e.g., the Zermelo numerals.
    [Show full text]