On numbers. A pearl seminar Mats Ehrnström
|
|
|
- Ambrose Nichols
- 5 years ago
- Views:
Transcription
1 On numbers A pearl seminar Mats Ehrnström
2
3 We actually need to count.
4 We actually need to count.
5 We actually need to count. But already know, without having counted all, many of us feel we have counted more than half.
6 Quantity perception the ability to discern ratios is innate in human beings and many animals. Infants typically succed below a 4:5 ratio (newborns lower).
7 Quantity perception the ability to discern ratios is innate in human beings and many animals. Infants typically succed below a 4:5 ratio (newborns lower). Subitizing is the ability to immediately and correctly determine low quantities (typically <5) without counting them. This universal ability does not limit itself to visual objects, but includes sounds and tactile perception.
8 Quantity perception the ability to discern ratios is innate in human beings and many animals. Infants typically succed below a 4:5 ratio (newborns lower). Subitizing is the ability to immediately and correctly determine low quantities (typically <5) without counting them. This universal ability does not limit itself to visual objects, but includes sounds and tactile perception.
9 But man counts. Typically, ordered sets are perceived as larger than unordered sets.
10 Peano s axioms But man counts.
11 Peano s axioms But man counts. A1. 1 is a natural number.
12 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural
13 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural A6. Every natural number n has a successor, S(n).
14 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural A6. Every natural number n has a successor, S(n). A7. 1 is the successor of no natural number.
15 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural A6. Every natural number n has a successor, S(n). A7. 1 is the successor of no natural number. A8. S(m) = S(n) => m=n
16 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural A6. Every natural number n has a successor, S(n). A7. 1 is the successor of no natural number. A8. S(m) = S(n) => m=n A9. Axiom of induction
17 Peano s axioms But man counts. A1. 1 is a natural number. A2-5. n=n; m=n => n=m; m=n, n=k => m=k; n natural, m=n => m natural A6. Every natural number n has a successor, S(n). A7. 1 is the successor of no natural number. A8. S(m) = S(n) => m=n A9. Axiom of induction
18 But man counts. The Lebombo bone the oldest known mathematical artefact
19 But man counts. The Lebombo bone the oldest known mathematical artefact Dated to B.C. Found in the Lebombo Mountains, Swaziland. Contains 29 (30?) notches carved into baboon bone.
20 But man counts. The Lebombo bone the oldest known mathematical artefact Dated to B.C. Found in the Lebombo Mountains, Swaziland. Contains 29 (30?) notches carved into baboon bone. A lunar cycle is days
21 The second oldest is the But man counts. Ishango bone.
22 The second oldest is the But man counts. Ishango bone. Older than years. Found near Virunga National Park, Democratic Republic of Congo. Contains three columns of grouped notches.
23 The second oldest is the But man counts. Ishango bone. 11, 13, 17, 19
24 The second oldest is the But man counts. Ishango bone. 11, 13, 17, 19 A prime quadruplet: p, p+2 p+6, p+8. Closest possible grouping for p>3. Proof: 0,2,4 = 0,2,1 (mod 3)
25 The second oldest is the But man counts. Ishango bone. Are there infinitely many prime quadruplets? 11, 13, 17, 19 Maybe. A prime quadruplet: p, p+2 p+6, p+8. Closest possible grouping for p>3. Proof: 0,2,4 = 0,2,1 (mod 3)
26 But of course, this might all be for the grip
27 So we learnt to count, and we learnt that there is no end of numbers.
28 So we learnt to count, and we learnt that there is no end of numbers. But what is?
29 So we learnt to count, and we learnt that there is no end of numbers. But what is? And how do we separate III IIII from III IIII?
30 So we learnt to count, and we learnt that there is no end of numbers. But what is? And how do we separate III IIII from III IIII? Or III III from III III?
31 We need a zero. And we need some signs. 0 +
32 We need a zero. And we need some signs. But are they really signs? 0 +
33 Of these, + is the easiest. Addition is the mapping N x N -> N, such that A 1: m+1 = S(m), A 2: m+s(n) = S(m+n). This reduces addition to counting. 0 +
34 0 is much worse. 0 +
35 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. 0 +
36 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder: 0 +
37 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder:
38 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder: Nothing: modern (Zermelo Fraenkel) set theory introduces zero as the empty set: 0={}
39 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder: Nothing: modern (Zermelo Fraenkel) set theory introduces zero as the empty set: 0={} 1=0[ {0} = {} [ {{}} = {{}} = {0}.
40 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder: Nothing: modern (Zermelo Fraenkel) set theory introduces zero as the empty set: 0={} 1=0[ {0} = {} [ {{}} = {{}} = {0}. 2=1[ {1} = {{}} [ { {{}} } = { {}, {{}} } = {0, 1}.
41 0 is much worse. It is a placeholder, a representation of nothing (i.e., of no thing ), and an additive identity. Placeholder: Nothing: modern (Zermelo Fraenkel) set theory introduces zero as the empty set: 0={} 1=0[ {0} = {} [ {{}} = {{}} = {0}. 2=1[ {1} = {{}} [ { {{}} } = { {}, {{}} } = {0, 1}. 3=2[ {2} =...= { {}, {{}}, {{{}}} } = {0, 1, 2}.
42 The strongest argument for that the notion of difficult, is probably history itself. 0 is 0
43 The strongest argument for that the notion of difficult, is probably history itself. Zero as a placeholder appears only 300 BC (~2000 years later than the decimal system and the first calculations of.) 0 is 0
44 The strongest argument for that the notion of difficult, is probably history itself. Zero as a placeholder appears only 300 BC (~2000 years later than the decimal system and the first calculations of.) 0 is Differing vague notions of zero as a symbol for nothing appearing from around 100 AD and 0 onwards (e.g. in Ptolemy s Almagest).
45 The strongest argument for that the notion of difficult, is probably history itself. Zero as a placeholder appears only 300 BC (~2000 years later than the decimal system and the first calculations of.) 0 is Differing vague notions of zero as a symbol for nothing appearing from around 100 AD and 0 onwards (e.g. in Ptolemy s Almagest). The use of zero as a number is generally credited 0 0 =0 to the indians. In 628 AD Brahmagupta published arithmetic rules for zero. The modern place value decimal system also appeared in India at that time.
46 The strongest argument for that the notion of difficult, is probably history itself. Zero as a placeholder appears only 300 BC (~2000 years later than the decimal system and the first calculations of.) 0 is Differing vague notions of zero as a symbol for nothing appearing from around 100 AD and 0 onwards (e.g. in Ptolemy s Almagest). The use of zero as a number is generally credited 0 0 =0 to the indians. In 628 AD Brahmagupta published arithmetic rules for zero. The modern place value decimal system also appeared in India at that time. Persian sifr (meaning empty ) for zero in 976 AD.
47 Still is worst of them all. It is the sign of both a binary and a unary operation, a part of the sign for additive inverses, and a notion that lacks (primitive) real world representations.
48 Still is worst of them all. It is the sign of both a binary and a unary operation, a part of the sign for additive inverses, and a notion that lacks (primitive) real world representations. - is a mapping, say, Z X Z -> Z (as in 0-3).
49 Still is worst of them all. It is the sign of both a binary and a unary operation, a part of the sign for additive inverses, and a notion that lacks (primitive) real world representations. - is a mapping, say, Z X Z -> Z (as in 0-3). - is a mapping Z -> Z (as in -3).
50 Still is worst of them all. It is the sign of both a binary and a unary operation, a part of the sign for additive inverses, and a notion that lacks (primitive) real world representations. - is a mapping, say, Z X Z -> Z (as in 0-3). - is a mapping Z -> Z (as in -3). - is part of the signs for Z_ (as in -3).
51 Still is worst of them all. It is the sign of both a binary and a unary operation, a part of the sign for additive inverses, and a notion that lacks (primitive) real world representations. - is a mapping, say, Z X Z -> Z (as in 0-3). - is a mapping Z -> Z (as in -3). - is part of the signs for Z_ (as in -3). And what is (0) - (3) = -(3) = (-3) anyway?
52 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5),
53 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5), [(m, n)] + [(k, l)] = [(m + k, n + l)]
54 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5), [(m, n)] + [(k, l)] = [(m + k, n + l)] [(m, n)] = [(n, m)]
55 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5), [(m, n)] + [(k, l)] = [(m + k, n + l)] [(m, n)] = [(n, m)] [(m, n)] [(k, l)] = [(m, n)] + [(l, k)]
56 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5), [(m, n)] + [(k, l)] = [(m + k, n + l)] [(m, n)] = [(n, m)] [(m, n)] [(k, l)] = [(m, n)] + [(l, k)] [(m, n)] [(k, l)] = [(mk + nl, ml + nk)]
57 And what is (0) - (3) = -(3) = (-3) anyway? -3 is the equivalence class of all pairs of natural numbers (m,n) of the form (0,3), (1,4), (2,5), [(m, n)] + [(k, l)] = [(m + k, n + l)] [(m, n)] = [(n, m)] [(m, n)] [(k, l)] = [(m, n)] + [(l, k)] [(m, n)] [(k, l)] = [(mk + nl, ml + nk)] But this took a long time...
58 The first recorded use of negative numbers is Chinese, from around 200 BC =0
59 The first recorded use of negative numbers is Chinese, from around 200 BC. But: With almost no exception, negative numbers has historically always been introduced and understood as debts or financial deficits =0
60 The first recorded use of negative numbers is Chinese, from around 200 BC. But: With almost no exception, negative numbers has historically always been introduced and understood as debts or financial deficits. There has been confusion in the separation of the operation and the elements =0
61 The first recorded use of negative numbers is Chinese, from around 200 BC. But: With almost no exception, negative numbers has historically always been introduced and understood as debts or financial deficits. There has been confusion in the separation of the operation and the elements. It is not clear when the number line (with negative numbers) was introduced, or if it is even appropriate for man s (or children s) understanding of numbers and quantities =0
62 The first recorded use of negative numbers is Chinese, from around 200 BC. But: With almost no exception, negative numbers has historically always been introduced and understood as debts or financial deficits. There has been confusion in the separation of the operation and the elements. It is not clear when the number line (with negative numbers) was introduced, or if it is even appropriate for man s (or children s) understanding of numbers and quantities =0 Negative integers (as numbers) were generally accepted by mathematicians only as the same time as the complex numbers. C
63
64 How do humans store or represent numbers and quantities?
65 How do humans store or represent numbers and quantities? Most of neuroscience is due to human disaster and tragedy. Phineas Gage ( )
66 How do humans store or represent numbers and quantities? Most of neuroscience is due to human disaster and tragedy. Phineas Gage ( )
67 How do humans store or represent numbers and quantities? Most of neuroscience is due to human disaster and tragedy. Phineas Gage ( )
68 How do humans store or represent numbers and quantities? Clear: there is a separate mental structure (not necessarily localised) for handling numbers. Signora Gaddi:
69 How do humans store or represent numbers and quantities? Clear: there is a separate mental structure (not necessarily localised) for handling numbers. Signora Gaddi: the ability to handle numbers is separate from the ability to handle logical propositions.
70 How do humans store or represent numbers and quantities? Clear: there is a separate mental structure (not necessarily localised) for handling numbers. Signora Gaddi: the ability to handle numbers is separate from the ability to handle logical propositions. handling large numbers is a mnemonic feat. It can be trained, but in some persons is due to synesthesia.
71 How do humans store or represent numbers and quantities? Clear: there is a separate mental structure (not necessarily localised) for handling numbers. Signora Gaddi: the ability to handle numbers is separate from the ability to handle logical propositions. handling large numbers is a mnemonic Daniel Tammet: feat. It can be trained, but in some persons is due to synesthesia.
72 How do humans store or represent numbers and quantities? To Tammet every integer has a specific Signora Gaddi: form, touch, smell, or other characteristic. handling large numbers is a mnemonic Daniel Tammet: feat. It can be trained, but in some persons is due to synesthesia.
73 =0 C
74 The concept of infinity (as such) is evidence at least from 300 B.C., both in European and Indian writings =0 C
75 The concept of infinity (as such) is evidence at least from 300 B.C., both in European and Indian writings. The concept of infinity (as innumerability) is probably innate in humans, and present as 1 soon as a child can count =0 C
76 The concept of infinity (as such) is evidence at least from 300 B.C., both in European and Indian writings. The concept of infinity (as innumerability) is probably innate in humans, and present as 1 soon as a child can count. Even though, in much of mathematics, the symbol 1 is not actually used for any object, but as a shorthand for limits, the =0 concept of infinite sets (and processes) is unavoidable. C
77 The concept of infinity (as such) is evidence at least from 300 B.C., both in European and Indian writings. The concept of infinity (as innumerability) is probably innate in humans, and present as 1 soon as a child can count. Even though, in much of mathematics, the symbol 1 is not actually used for any object, but as a shorthand for limits, the =0 concept of infinite sets (and processes) is unavoidable. which leads us to Cantor, Russell, Gödel, Cohen, and the continuum hypothesis C
78 In the late 19th century several mathematicians started rejecting the idea (from Galileo and Euclid) that the whole must be larger than any one part of it.
79 In the late 19th century several mathematicians started rejecting the idea (from Galileo and Euclid) that the whole must be larger than any one part of it.
80 In the late 19th century several mathematicians started rejecting the idea (from Galileo and Euclid) that the whole must be larger than any one part of it. Most famously, Cantor, using the concept of cardinality, that (i) A = B <=> A is a bijective image of B, (ii) Each cardinality has a representative set, showed that the powerset (set of all subsets) of any set always has a greater cardinality than the set itself.
81 In the late 19th century several mathematicians started rejecting the idea (from Galileo and Euclid) that the whole must be larger than any one part of it. Most famously, Cantor, using the concept of cardinality, that (i) A = B <=> A is a bijective image of B, (ii) Each cardinality has a representative set, showed that the powerset (set of all subsets) of any set always has a greater cardinality than the set itself. N 0 1 (size of the set of all countable ordinals) R 0 (size of the powerset of N)
82 The continuum 1 0?
83 The continuum 1 0? But a number of obstacles arose:
84 The continuum 1 0? But a number of obstacles arose: 1901 Russell s paradox R = {x 62 x} =) [R 2 R () R 62 R]
85 The continuum 1 0? But a number of obstacles arose: 1901 Russell s paradox R = {x 62 x} =) [R 2 R () R 62 R] Saved by Zermelo Fraenkel set theory (ZFC).
86 The continuum 1 0? But a number of obstacles arose: 1901 Russell s paradox R = {x 62 x} =) [R 2 R () R 62 R] Saved by Zermelo Fraenkel set theory (ZFC) Gödel s first incompleteness theorem: A formal system with an inductively enumerable set of axioms capable of expressing fundamental (Peano) arithmetic cannot be both complete and consistent.
87 The continuum 1 0? But a number of obstacles arose: 1901 Russell s paradox R = {x 62 x} =) [R 2 R () R 62 R] Saved by Zermelo Fraenkel set theory (ZFC) Gödel s first incompleteness theorem: A formal system with an inductively enumerable set of axioms capable of expressing fundamental (Peano) arithmetic cannot be both complete and consistent Gödel shows that the continuum hypothesis cannot be disproved by means of ZFC.
88 The continuum 1 0? But a number of obstacles arose: 1901 Russell s paradox R = {x 62 x} =) [R 2 R () R 62 R] Saved by Zermelo Fraenkel set theory (ZFC) Gödel s first incompleteness theorem: A formal system with an inductively enumerable set of axioms capable of expressing fundamental (Peano) arithmetic cannot be both complete and consistent Gödel shows that the continuum hypothesis cannot be disproved by means of ZFC Cohen shows that the continuum hypothesis cannot be proved by means of ZFC.
89 One can assume, or not assume, the existence of a set of cardinality strictly between that of the natural numbers and that of the reals, and do mathematics with or without this assumption and there is no logical argument against it.
90 One can assume, or not assume, the existence of a set of cardinality strictly between that of the natural numbers and that of the reals, and do mathematics with or without this assumption and there is no logical argument against it. Still Some believe the continuum hypothesis is false; others believe it is true.
91 One can assume, or not assume, the existence of a set of cardinality strictly between that of the natural numbers and that of the reals, and do mathematics with or without this assumption and there is no logical argument against it. Still Some believe the continuum hypothesis is false; others believe it is true. Some believe ZFC lacks some important axiom.
92 One can assume, or not assume, the existence of a set of cardinality strictly between that of the natural numbers and that of the reals, and do mathematics with or without this assumption and there is no logical argument against it. Still Some believe the continuum hypothesis is false; others believe it is true. Some believe ZFC lacks some important axiom. And some think it doesn t have to do with mathematics.
93 One can assume, or not assume, the existence of a set of cardinality strictly between that of the natural numbers and that of the reals, and do mathematics with or without this assumption and there is no logical argument against it. Still Some believe the continuum hypothesis is false; others believe it is true. Some believe ZFC lacks some important axiom. And some think it doesn t have to do with mathematics. But possibly more disturbing is the fact that this might be just one of many propositions that doesn t have to do with mathematics.
94 To put it short: Today s concepts of numbers and arithmetics have taken a long time to understand and develop.
95 To put it short: Today s concepts of numbers and arithmetics have taken a long time to understand and develop. Our everyday use and understanding of these might not always correspond to either their historical or current mathematical meaning.
96 To put it short: Today s concepts of numbers and arithmetics have taken a long time to understand and develop. Our everyday use and understanding of these might not always correspond to either their historical or current mathematical meaning. Although the level of abstraction has no end, there is a clear and universal level of when we need to make an effort, and when not
97 To put it short: Today s concepts of numbers and arithmetics have taken a long time to understand and develop. Our everyday use and understanding of these might not always correspond to either their historical or current mathematical meaning. Although the level of abstraction has no end, there is a clear and universal level of when we need to make an effort, and when not and that is by 4.
98
Part IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
Set Theory and the Foundation of Mathematics. June 19, 2018
1 Set Theory and the Foundation of Mathematics June 19, 2018 Basics Numbers 2 We have: Relations (subsets on their domain) Ordered pairs: The ordered pair x, y is the set {{x, y}, {x}}. Cartesian products
Axiomatic set theory. Chapter Why axiomatic set theory?
Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its
Introduction to Logic and Axiomatic Set Theory
Introduction to Logic and Axiomatic Set Theory 1 Introduction In mathematics, we seek absolute rigor in our arguments, and a solid foundation for all of the structures we consider. Here, we will see some
Products, Relations and Functions
Products, Relations and Functions For a variety of reasons, in this course it will be useful to modify a few of the settheoretic preliminaries in the first chapter of Munkres. The discussion below explains
INFINITY: CARDINAL NUMBERS
INFINITY: CARDINAL NUMBERS BJORN POONEN 1 Some terminology of set theory N := {0, 1, 2, 3, } Z := {, 2, 1, 0, 1, 2, } Q := the set of rational numbers R := the set of real numbers C := the set of complex
Gödel s Proof. Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College. Kurt Gödel
Gödel s Proof Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College Kurt Gödel 24.4.06-14.1.78 1 ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS 11 by Kurt Gödel,
We introduce one more operation on sets, perhaps the most important
11. The power set Please accept my resignation. I don t want to belong to any club that will accept me as a member. Groucho Marx We introduce one more operation on sets, perhaps the most important one:
Hilbert s problems, Gödel, and the limits of computation
Hilbert s problems, Gödel, and the limits of computation Logan Axon Gonzaga University April 6, 2011 Hilbert at the ICM At the 1900 International Congress of Mathematicians in Paris, David Hilbert gave
Informal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
CITS2211 Discrete Structures (2017) Cardinality and Countability
CITS2211 Discrete Structures (2017) Cardinality and Countability Highlights What is cardinality? Is it the same as size? Types of cardinality and infinite sets Reading Sections 45 and 81 84 of Mathematics
Hence, the sequence of triangular numbers is given by., the. n th square number, is the sum of the first. S n
Appendix A: The Principle of Mathematical Induction We now present an important deductive method widely used in mathematics: the principle of mathematical induction. First, we provide some historical context
Set Theory History. Martin Bunder. September 2015
Set Theory History Martin Bunder September 2015 What is a set? Possible Definition A set is a collection of elements having a common property Abstraction Axiom If a(x) is a property ( y)( x)(x y a(x))
2.2 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
INCOMPLETENESS I by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Invitation
INCOMPLETENESS I by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Invitation to Mathematics Series Department of Mathematics Ohio
Philosophies of Mathematics. The Search for Foundations
Philosophies of Mathematics The Search for Foundations Foundations What are the bedrock, absolutely certain, immutable truths upon which mathematics can be built? At one time, it was Euclidean Geometry.
MATH 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
Victoria Gitman and Thomas Johnstone. New York City College of Technology, CUNY
Gödel s Proof Victoria Gitman and Thomas Johnstone New York City College of Technology, CUNY vgitman@nylogic.org http://websupport1.citytech.cuny.edu/faculty/vgitman tjohnstone@citytech.cuny.edu March
Notes on induction proofs and recursive definitions
Notes on induction proofs and recursive definitions James Aspnes December 13, 2010 1 Simple induction Most of the proof techniques we ve talked about so far are only really useful for proving a property
Axioms for Set Theory
Axioms for Set Theory The following is a subset of the Zermelo-Fraenkel axioms for set theory. In this setting, all objects are sets which are denoted by letters, e.g. x, y, X, Y. Equality is logical identity:
MAGIC Set theory. lecture 2
MAGIC Set theory lecture 2 David Asperó University of East Anglia 22 October 2014 Recall from last time: Syntactical vs. semantical logical consequence Given a set T of formulas and a formula ', we write
Lecture Notes on Discrete Mathematics. October 15, 2018 DRAFT
Lecture Notes on Discrete Mathematics October 15, 2018 2 Contents 1 Basic Set Theory 5 1.1 Basic Set Theory....................................... 5 1.1.1 Union and Intersection of Sets...........................
MAGIC Set theory. lecture 1
MAGIC Set theory lecture 1 David Asperó University of East Anglia 15 October 2014 Welcome Welcome to this set theory course. This will be a 10 hour introduction to set theory. The only prerequisite is
Farewell (for now) to modular arithmetic...
Today. Farewell to modular arithmetic. Until the midterm. And final. Coutability and Uncountability. Undecidability. How big are the reals or the integers? Farewell (for now) to modular arithmetic... Modular
A BRIEF INTRODUCTION TO ZFC. Contents. 1. Motivation and Russel s Paradox
A BRIEF INTRODUCTION TO ZFC CHRISTOPHER WILSON Abstract. We present a basic axiomatic development of Zermelo-Fraenkel and Choice set theory, commonly abbreviated ZFC. This paper is aimed in particular
Cantor and sets: La diagonale du fou
Judicaël Courant 2011-06-17 Lycée du Parc (moving to Lycée La Martinière-Monplaisir) Outline 1 Cantor s paradise 1.1 Introduction 1.2 Countable sets 1.3 R is not countable 1.4 Comparing sets 1.5 Cardinals
INDEPENDENCE OF THE CONTINUUM HYPOTHESIS
INDEPENDENCE OF THE CONTINUUM HYPOTHESIS CAPSTONE MATT LUTHER 1 INDEPENDENCE OF THE CONTINUUM HYPOTHESIS 2 1. Introduction This paper will summarize many of the ideas from logic and set theory that are
Hilbert s problems, Gödel, and the limits of computation
Hilbert s problems, Gödel, and the limits of computation Logan Axon Gonzaga University November 14, 2013 Hilbert at the ICM At the 1900 International Congress of Mathematicians in Paris, David Hilbert
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
Extended Essay - Mathematics
Extended Essay - Mathematics Creating a Model to Separate or Group Number Sets by their Cardinalities Pope John Paul II C.S.S. September 2009 Candidate Number: 001363-012 The conquest of the actual infinite
RESOLVING RUSSELL S PARADOX WITHIN CANTOR S INTUITIVE SET THEORY. Feng Xu ( Abstract
RESOLVING RUSSELL S PARADOX WITHIN CANTOR S INTUITIVE SET THEORY Feng Xu (e-mail: xtwan@yahoo.com) Abstract The set of all the subsets of a set is its power set, and the cardinality of the power set is
Gödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science
www.logicnest.com Gödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science The Hume Cantor Principle: If there is a 1-1 correspondence between two collections,
1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
Today. Wrapup of Polynomials...and modular arithmetic. Coutability and Uncountability.
Today. Wrapup of Polynomials...and modular arithmetic. Coutability and Uncountability. Reed-Solomon code. Problem: Communicate n packets m 1,...,m n on noisy channel that corrupts k packets. Reed-Solomon
The Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018)
The Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018) From the webpage of Timithy Kohl, Boston University INTRODUCTION Note. We will consider infinity from two different perspectives:
Lecture 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,
Tutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
Obsevations, Truth and Logic
Obsevations, Truth and Logic Jaap van Oosten Department of Mathematics Utrecht University Axioma, Groningen May 9, 2018 Art and Science have the same basic subject matter: Observations. Art (visual art):
MATH 13 FINAL EXAM SOLUTIONS
MATH 13 FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers. T F
Math 144 Summer 2012 (UCR) Pro-Notes June 24, / 15
Before we start, I want to point out that these notes are not checked for typos. There are prbally many typeos in them and if you find any, please let me know as it s extremely difficult to find them all
PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms
PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns
SOME TRANSFINITE INDUCTION DEDUCTIONS
SOME TRANSFINITE INDUCTION DEDUCTIONS SYLVIA DURIAN Abstract. This paper develops the ordinal numbers and transfinite induction, then demonstrates some interesting applications of transfinite induction.
The natural numbers. The natural numbers come with an addition +, a multiplication and an order < p < q, q < p, p = q.
The natural numbers N = {0, 1,, 3, } The natural numbers come with an addition +, a multiplication and an order < p, q N, p + q N. p, q N, p q N. p, q N, exactly one of the following holds: p < q, q
Grade 7/8 Math Circles Winter March 20/21/22 Types of Numbers
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Winter 2018 - March 20/21/22 Types of Numbers Introduction Today, we take our number
Chapter 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
1 of 8 7/15/2009 3:43 PM Virtual Laboratories > 1. Foundations > 1 2 3 4 5 6 7 8 9 6. Cardinality Definitions and Preliminary Examples Suppose that S is a non-empty collection of sets. We define a relation
Algorithms: Lecture 2
1 Algorithms: Lecture 2 Basic Structures: Sets, Functions, Sequences, and Sums Jinwoo Kim jwkim@jjay.cuny.edu 2.1 Sets 2 1 2.1 Sets 3 2.1 Sets 4 2 2.1 Sets 5 2.1 Sets 6 3 2.1 Sets 7 2.2 Set Operations
König s Lemma and Kleene Tree
König s Lemma and Kleene Tree Andrej Bauer May 3, 2006 Abstract I present a basic result about Cantor space in the context of computability theory: the computable Cantor space is computably non-compact.
Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Set Theory Zak Mesyan University of Colorado Colorado Springs The Real Numbers In the 19th century attempts to prove facts about the real numbers were limited by the lack of a rigorous
Great Theoretical Ideas
15-251 Great Theoretical Ideas in Computer Science Gödel s Legacy: Proofs and Their Limitations Lecture 25 (November 16, 2010) The Halting Problem A Quick Recap of the Previous Lecture Is there a program
Sets and Infinity. James Emery. Edited: 2/25/ Cardinal Numbers 1. 2 Ordinal Numbers 6. 3 The Peano Postulates for the Natural Numbers 7
Sets and Infinity James Emery Edited: 2/25/2017 Contents 1 Cardinal Numbers 1 2 Ordinal Numbers 6 3 The Peano Postulates for the Natural Numbers 7 4 Metric Spaces 8 5 Complete Metric Spaces 8 6 The Real
0.Axioms for the Integers 1
0.Axioms for the Integers 1 Number theory is the study of the arithmetical properties of the integers. You have been doing arithmetic with integers since you were a young child, but these mathematical
Basic set-theoretic techniques in logic Part III, Transfinite recursion and induction
Basic set-theoretic techniques in logic Part III, Transfinite recursion and induction Benedikt Löwe Universiteit van Amsterdam Grzegorz Plebanek Uniwersytet Wroc lawski ESSLLI 2011, Ljubljana, Slovenia
INDUCTION AND RECURSION
INDUCTION AND RECURSION Jorma K. Mattila LUT, Department of Mathematics and Physics 1 Induction on Basic and Natural Numbers 1.1 Introduction In its most common form, mathematical induction or, as it is
WHY ISN T THE CONTINUUM PROBLEM ON THE MILLENNIUM ($1,000,000) PRIZE LIST?
WHY ISN T THE CONTINUUM PROBLEM ON THE MILLENNIUM ($1,000,000) PRIZE LIST? Solomon Feferman CSLI Workshop on Logic, Rationality and Intelligent Interaction Stanford, June 1, 2013 Why isn t the Continuum
Unit 3: Definitions, Theorems and Proofs
LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22A Unit 3: Definitions, Theorems and Proofs Seminar 3.1. Theorems are mathematical statements which can be verified using proofs. Theorems are the backbone of mathematics.
NUMBERS It s the numbers that count
NUMBERS It s the numbers that count Starting from the intuitively obvious this note discusses some of the perhaps not so intuitively obvious aspects of numbers. Henry 11/1/2011 NUMBERS COUNT! Introduction
Equivalent Forms of the Axiom of Infinity
Equivalent Forms of the Axiom of Infinity Axiom of Infinity 1. There is a set that contains each finite ordinal as an element. The Axiom of Infinity is the axiom of Set Theory that explicitly asserts that
2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
This section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.
1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as
Lecture 2: What is Proof?
Lecture 2: What is Proof? Math 295 08/26/16 Webster Proof and Its History 8/2016 1 / 1 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct
Definition: 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
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness
Philosophy of Religion. Notes on Infinity
Notes on Infinity Infinity in Classical and Medieval Philosophy Aristotle (Metaphysics 986 a 22) reports that limited (peras, πέρας) and unlimited or infinite (apeiron, ἄπειρον) occur as the first pairing
Hilbert and the concept of axiom
Hilbert and the concept of axiom Giorgio Venturi Scuola Normale Superiore di Pisa Giorgio Venturi (SNS) Hilbert and the concept of axiom 1/24 First period Axiomatic method in the first period The actual
SETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
Section 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
CS 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
ABOUT THE CLASS AND NOTES ON SET THEORY
ABOUT THE CLASS AND NOTES ON SET THEORY About the Class Evaluation. Final grade will be based 25%, 25%, 25%, 25%, on homework, midterm 1, midterm 2, final exam. Exam dates. Midterm 1: Oct 4. Midterm 2:
Paradox and Infinity Problem Set 6: Ordinal Arithmetic
24.118 Paradox and Infinity Problem Set 6: Ordinal Arithmetic You will be graded both on the basis of whether your answers are correct and on the basis of whether they are properly justified. There is
Meta-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
Chapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known
Herbrand 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
INTRODUCTION TO CARDINAL NUMBERS
INTRODUCTION TO CARDINAL NUMBERS TOM CUCHTA 1. Introduction This paper was written as a final project for the 2013 Summer Session of Mathematical Logic 1 at Missouri S&T. We intend to present a short discussion
Chapter 0. Introduction: Prerequisites and Preliminaries
Chapter 0. Sections 0.1 to 0.5 1 Chapter 0. Introduction: Prerequisites and Preliminaries Note. The content of Sections 0.1 through 0.6 should be very familiar to you. However, in order to keep these notes
To Infinity and Beyond
University of Waterloo How do we count things? Suppose we have two bags filled with candy. In one bag we have blue candy and in the other bag we have red candy. How can we determine which bag has more
Cardinality of Sets. P. Danziger
MTH 34-76 Cardinality of Sets P Danziger Cardinal vs Ordinal Numbers If we look closely at our notions of number we will see that in fact we have two different ways of conceiving of numbers The first is
Notes 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,...}
CSE 20 Discrete Math. Winter, January 24 (Day 5) Number Theory. Instructor: Neil Rhodes. Proving Quantified Statements
CSE 20 Discrete Math Proving Quantified Statements Prove universal statement: x D, P(x)Q(x) Exhaustive enumeration Generalizing from the generic particular Winter, 2006 Suppose x is in D and P(x) Therefore
(1.3.1) and in English one says a is an element of M. The statement that a is not an element of M is written as a M
1.3 Set Theory I As long as the terms of a mathematical theory are names of concrete objects as concrete as mothers breast, the very first object that received a name in human languages - there is not
Sets, Models and Proofs. I. Moerdijk and J. van Oosten Department of Mathematics Utrecht University
Sets, Models and Proofs I. Moerdijk and J. van Oosten Department of Mathematics Utrecht University 2000; revised, 2006 Contents 1 Sets 1 1.1 Cardinal Numbers........................ 2 1.1.1 The Continuum
Bounded Infinite Sequences/Functions : Orders of Infinity
Bounded Infinite Sequences/Functions : Orders of Infinity by Garimella Ramamurthy Report No: IIIT/TR/2009/247 Centre for Security, Theory and Algorithms International Institute of Information Technology
MATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
Reed-Solomon code. P(n + 2k)
Reed-Solomon code. Problem: Communicate n packets m 1,...,m n on noisy channel that corrupts k packets. Reed-Solomon Code: 1. Make a polynomial, P(x) of degree n 1, that encodes message: coefficients,
The Reflection Theorem
The Reflection Theorem Formalizing Meta-Theoretic Reasoning Lawrence C. Paulson Computer Laboratory Lecture Overview Motivation for the Reflection Theorem Proving the Theorem in Isabelle Applying the Reflection
Chapter 4. Basic Set Theory. 4.1 The Language of Set Theory
Chapter 4 Basic Set Theory There are two good reasons for studying set theory. First, it s a indispensable tool for both logic and mathematics, and even for other fields including computer science, linguistics,
Arithmetic and Incompleteness. Will Gunther. Goals. Coding with Naturals. Logic and Incompleteness. Will Gunther. February 6, 2013
Logic February 6, 2013 Logic 1 2 3 Logic About Talk Logic Things talk Will approach from angle of computation. Will not assume very much knowledge. Will prove Gödel s Incompleteness Theorem. Will not talk
Trees and generic absoluteness
in ZFC University of California, Irvine Logic in Southern California University of California, Los Angeles November 6, 03 in ZFC Forcing and generic absoluteness Trees and branches The method of forcing
Mathematical structures
Mathematical structures Lund 2015 Preface I still remember a guy sitting on a couch, thinking very hard, and another guy standing in front of him, saying, And therefore such-and-such is true. Why is that?
CSE 311: Foundations of Computing. Lecture 26: Cardinality
CSE 311: Foundations of Computing Lecture 26: Cardinality Cardinality and Computability Computers as we know them grew out of a desire to avoid bugs in mathematical reasoning A brief history of reasoning
Math 109 September 1, 2016
Math 109 September 1, 2016 Question 1 Given that the proposition P Q is true. Which of the following must also be true? A. (not P ) or Q. B. (not Q) implies (not P ). C. Q implies P. D. A and B E. A, B,
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity W. Hugh Woodin Harvard University January 9, 2017 V : The Universe of Sets The power set Suppose X is a set. The powerset of X
A SYSTEM OF AXIOMATIC SET THEORY PART VI 62
THE JOOBNAL OF SYMBOLIC LOGIC Volume 13, Number 2, June 1948 A SYSTEM OF AXIOMATIC SET THEORY PART VI 62 PAUL BEKNAYS 16. The r61e of the restrictive axiom. Comparability of classes. Till now we tried
Gödel s Programm and Ultimate L
Gödel s Programm and Ultimate L Fudan University National University of Singapore, September 9, 2017 Outline of Topics 1 CT s Problem 2 Gödel s Program 3 Ultimate L 4 Conclusion Remark Outline CT s Problem
THE MATHEMATICAL UNIVERSE IN A NUTSHELL
THE MATHEMATICAL UNIVERSE IN A NUTSHELL K. K. NAMBIAR ABSTRACT. The mathematical universe discussed here gives models of possible structures our physical universe can have. Keywords Cosmology, White Hole,
Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)
http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product
Now, with all the definitions we ve made, we re ready see where all the math stuff we took for granted, like numbers, come from.
The Natural Numbers Now, with all the definitions we ve made, we re ready see where all the math stuff we took for granted, like numbers, come from. Math 144 Summer 2012 (UCR) Pro-Notes July 6, 2012 1
Descriptive Set Theory: Why Should We Study It?
Descriptive Set Theory: Why Should We Study It? Thomas D. Gilton 1 Department of Mathematics College of Arts & Sciences Faculty Mentor: Mentor s Department: Mentor s College: Dr. John Krueger Mathematics
Introduction to Proofs
Introduction to Proofs Notes by Dr. Lynne H. Walling and Dr. Steffi Zegowitz September 018 The Introduction to Proofs course is organised into the following nine sections. 1. Introduction: sets and functions
Russell s logicism. Jeff Speaks. September 26, 2007
Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................