Set Theory Overview
Set theory is one of the fundamental fields of mathematics, which deals with the concept of sets. A set is a collection of elements, or mathematically, a collection of objects that satisfy certain conditions.
A set may be expressed in various forms. For example, one can list a sequence of elements enclosed in {}, or list elements that satisfy certain conditions. For example, the set {1, 2, 3, 4} represents a collection consisting of the elements 1, 2, 3, 4, and the set {x | x is a natural number and x is even} represents a collection consisting of elements that are natural and even.
In set theory, the following concepts are mainly dealt with
- Element: refers to the individual objects that make up a set.
- Inclusion relation: denotes that a set contains all the elements of another set. For example, {1, 2} is a subset of {1, 2, 3}.
- Sum: Represents a set that contains all the elements of two or more sets. For example, the union of {1, 2} and {2, 3} is {1, 2, 3}.
- Product set: Represents a set consisting of elements common to two or more sets. For example, the product set of {1, 2} and {2, 3} is {2}.
- Difference set: The set consisting of one set minus the elements contained in another set. For example, the difference set obtained by removing {2, 3} from {1, 2, 3} is {1}.
- Inclusive Notation: This is a method of expressing a set of elements that satisfy certain conditions. For example, the set {x | x is a natural number and x is even} represents a collection of elements that are natural and even.
Set theory also plays an important role in other areas of mathematics. For example, the concepts of set theory are needed in fields such as topology, algebra, and foundations of mathematics. Furthermore, the concept of set theory is also used in fields such as information science and philosophy.
The first reference book is “What is a set? This book is an introductory book on set theory.
What role do sets play in modern mathematics?Modern mathematics cannot develop without sets. The question of what is a set is the deepest question in modern mathematics, involving the search for new set axioms. This is a reissue of the famous book that explains the profound mysteries of the set concept and the romantic spirit of creativity hidden in set theory to people who are not trained in mathematics, with the addition of Cantor’s biography.
Chapter 1: Transformation of Positions Sets as a Transliteration Language
memo
P(x) x is in the set of those satisfying property P
P(a) : Determining P and a determines whether they are correct or incorrect
A collection of things satisfying property P: {x|P(x)}
a belongs to the set {x|P(x)} : a ∈ {x|P(x)}
Transformation of standpoints
Subject and predicate
The World of Simple and Clear Logic
Logical operations
Sets as Translation Terms
Common parts and union sets
Negation Translation
Difference of sets
All and existence
Translation of ∀ and ∃
Subset
The Empty Set
Chapter 2: Creation, Exile from Paradise
memo
A set of collections of things
node?
b∈{a1,a2,.... . an}=a1∨b=a2∨b...∨b=an
A collection of things satisfying a property
Edge?
All subsets of A-{1,2,3...,N}
When n=3
(a1+b1)(a2+b2)(a3+b3)
Let a1,a2,a3 be "1 in", "2 in", and "3 in".
Let b1,b2,b3 be "1 is not contained", "2 is not contained", and "3 is not contained".
a1a2a3 is {1,2,3}.
a1b2a3 is {1,3}
a1a2b3 is {1,2}
{1,2...n} all subsets
(a1+b1)(a2+b2)...(an+bn)
Concentration of sets
Sets A and B have the same concentration = there is a one-to-one correspondence between A and B
Concentration = number of elements
When there is a one-to-one correspondence with a set of natural numbers, it is said to be "divisible" or "additive
Product set
The set consisting of all subsets of the set A as a whole
Genesis
Beginning of a person
Concentration of a set
Empty sets and subsets
Number of subsets
Concentration of sets
The product set
Cantors diagonal argument
The World of Sets
Expulsion from Paradise
Foundations of Mathematics
Chapter 3 Axiomatic Set Theory: Foundations of Modern Mathematics
MEMO
Zermelo's Set Theory
Function
Let f be a function from a to B
f:a→b
<x,y>∈f : x is mapped to y by f
Selection Axiom
There exists a selection function for any set
Frenkel's substitution axiom
ZF set theory (Zermelo-Frenkel)
Von Neumann's regularity axiom
BG set theory
Zermelo set theory
On Functions
Selection Axiom
Frenkel's Substitution Axiom
Von Neumann's regularity axiom
Axiomatic Set Theory
BG set theory
Chapter 4: Modern Set Theory: A Brilliant Development
MEMO
Cohen's work
Think of a space that is not just a set, but a space that contains various structures and operations defined on its components.
What is partially based on a and partially based on b?
Consider three different a,b,c spaces
[a∈u]=A
Intuitionistic Logic
Quantum Logic
Relation to topological spaces
Grothendieck
Cantor's set theory
Continuum hypothesis
Goeter's constructive set
Cohen's work
Unreachable number
Measurable number
Axiom of determination
Arabesque
Chapter 5: Invitation to the Future From My Standpoint
What is a set?
On Continuum
Cantor
Background of my life
Cantor's Set Theory
Disillusionment
Back to mathematics
On the contradictions of set theory
memo
What is a set?
A set is a collection of things considered as a whole
A "thing" is called an element
An element belonging to a set is represented by a symbol ∈.
Extensional notation" is a way of enumerating elements
Inclusive notation" is a way to write a definition in terms of conditions
axiomatic set theory" is a way to define what a set is with restrictions
Example: Binding to avoid Russell's paradox
Zermero-Fraenkel axiom (ZF) plus choice axiom (C) "ZFC axiom system" is the mainstream set theory system
Naive set theory", a set theory that does not go into ZFC axioms
Proposition
A proposition is a thing whose truth or falsity is known.
A ⇒ B
the set
B ⊂ A
A ∩ B
A ∪ B union set
A - B Difference set
Ac Complement set
Direct product set
Two things a and b are called an "ordered pair" made from a and b if (a, b) is an ordered pair made from a and b
Let A and B be sets; the set consisting of the entire ordered pair (a, b) constructed from a ∈ A and b ∈ B is called the "direct product" or "direct product set" of A and B and denoted by "A X B".
Power set
The set of all subsets of a set A is called the "power set" of A
Binary operations
Addition of integers is explained using mapping
Binomial operation: a generalization of the "rule for determining a new number from two numbers
μ:AxA → A:(x,y) → μ(x,y)
f(a, b) = a △ b : △ (arbitrary) symbol for the image of a binary operation
map
mapping
Let A and B be sets.
For each element of A, when one element of B is fixed
This correspondence is called a "map" from A to B
f : A → B
A: Domain
B: range of values
Necessary condition for determining f : A → B
(1) f(a) is fixed for any a ∈ A
However, if a ∈ A is not unique in the way it is described, the same source must correspond to any description
(2) f(a) determined by (1) is the source of B
Composite map
Let f : A → B and g : B → C be maps respectively
Composite maps g ⚪︎ f :A → C (a ↦ g( f(a)))
Restriction map
Consider the map f : A → B
Let C ⊂ A.
Since c ∈ C is also the source of A, f(c) ∈ B is determined
Then the map C → B is called a restriction of f to itself (restriction map), and we write f|C
Total projection
Consider the map f : A → B
Let f(C) ={ F(c) | c ∈ C} for C ⊂ A. This is called the image of C by f.
Since C is a subset of A and not the source, it is different from f(C) up to now.
In particular, when C=A, f(A) is called the image of f, which is also written as Imf.
The f for which Imf =B holds is called a surjection.
monjection
Consider the map f : A → B
f-1(b) = { a ∈ A | f(a) =b} for b ∈ B. This is called the "inverse image" of b by f.
f-1(C) = { a ∈ A | f(a) ∈ C} for C ⊂ B is called the inverse image of C by f.
For any b ∈ B, f is called an injection if f-1(b) contains at most one atom or no atoms.
All bijections (bijections)
F is both an all-injection and a bijection
Relation
A "relation" between two originals belonging to the same set
Relation
Let A be a set
A subset R of the direct product set A x A is called a "binary relation" or simply a "relation" on A.
When we want to make it clear that a relation R is defined on A, we write (A, R).
When R is a relation, we also write xRy to indicate that (x,y) ∈ R.
Ordered relations
A relation ⪯ on a set A is an "order relation (order)" or simply
(1) "reflexive law" x ⪯ x for any x ∈ A
(2) "transitive law" x ⪯ z if x ⪯ y, y ⪯ z
(3) If "asymmetric law" x ⪯ y, y ⪯ x then x = y
(A, ⪯) is called an "order set
Total order
If x⪯y or y⪯x holds for any two elements x, y ∈ A of an order set (A, ⪯), this order is called "total order" and this order set is called "totally order set.
Partial order
When we want to distinguish a partial order from a total order, we call it a "partial order
Order subset
Consider an ordered set (A, ⪯)
For B ⊂ A, if the order of B is defined by A, then B is also an ordered set
Let x be the maximal principal of A
For an ordered set (A, ⪯), x = y if x⪯y
Let x be the minimal source of A
For an element x of the ordered set (A, ⪯), if y⪯x then x=y
x is the maximal source of A
For any y ∈ A, if y⪯x then
x is the minimal source of A
If x⪯y for any y ∈ A
x is the upper bound of B
If x ∈ A satisfies y≼x (∀y ∈ B) for any subset B of the ordered set (A, ≼), then x is called the upper bound of B
B is bounded above.
B if there exists an upper bound on B
order ≼ on a set A is a well-ordered
The existence of a minimal source in any nonempty subset
Lexicographic order
Order X = N × N as follows
(1) If a0 =a1, then (a0,b0)≤(a1,b1) if b0 ≤b1.
(2) If a0 ≤a1 when a0 ̸=a1, then (a0,b0)≤(a1,b1).
Example
The following is a concrete example that may be somewhat difficult to understand. For simplicity, let < denote that (a0,b0)≤(a1,b1) or (a0,b0)̸=(a1,b1). (1, 1) < (1, 2) < (1, 3) < - - - - < (2, 1) < (2, 2) < (2, 3) < - - - - < (3, 1) < - - - - You will also notice that this is similar to the word order in the dictionary.
Inductive order
4.2.19 (inductive order). An order ≼ on a set A is an inductive order if any nonempty whole-order subset of A is bounded above. In this case, A is called an inductively ordered set.
Mathematical Induction and Transfinite Induction
Example
Mathematical induction
The proposition about the natural number n is true if (1) 1. (2) It is true for n if it is true for all natural numbers smaller than n. (3) is true for any n if and only if (2) is true for n if (2') is true for n-1. (2) can also be thought of in the form (2') If it is correct for n-1, it is also correct for n.
Transfinite induction
This generalizes to aligned sets. That is, if A is an ordered set, then the proposition for a ∈ A is (1) true for the least element of A. (2) If it is true for all elements smaller than a with respect to this ordering, then it is also true for a. is true for any a if and only if This is due to the fact that there is no infinite narrowly monotonically decreasing sequence in the ordered set. That is, if we determine a ∈ A, then the narrowly monotone decreasing sequence reaches its minimum source in a finite number of times. Therefore, the proposition is proved in a finite number of procedures. The generalization of mathematical induction to the set of sequences is called transfinite induction.
equivalence relation and analogy
Equivalence relation
A relation ∼ on a set A is an equivalence relation if it satisfies the following conditions
(1) [Reflection law] For any x ∈ A, x ∼ x
(2) [Symmetry law] If x ∼ y, then y ∼ x
(3) [Transitive law] x ∼ z if x ∼ y, y ∼ z
The equivalence relation is a mathematical formulation of the concept of "same
(1) The self-same element is "the same" as itself. (2) If x and y are "the same," then y and x are "the same. (3) If x and y are the same and y and z are the same, then x and z are the same.
Congruence of Integers
Difficulties
Concentration of sets
If a set S is a finite set, the number of elements in S is called the concentration of S.
Selection Axiom, Alignability Theorem, Zorn's Corollary
The next reference book is “Introduction to Set Theory.
This book is also an introduction to set theory.
The simple concepts of “a collection of things” and “continuity. After Cantor laid the foundations of set theory in the 19th century, Russell and various mathematicians discovered paradoxes and difficulties. Since then, set theory has made great progress, and its basic concepts are now indispensable not only to modern mathematics but also to philosophy, which makes full use of logic. This book explains the foundations of classical set theory in three parts: “Algebra of Sets,” “Concentration,” and “Ordered Numbers. The book is compact, yet carefully written, and is ideal for self-study. A well-established introduction to the subject by the author of Introduction to Mathematics.”
Introduction to Set Theory Math & Science Aka Setsuya
On the occasion of the paperback edition
This book is a detailed exposition of the "classical" set theory originated by Georg Cantor.
It only touches a little on "axiomatic" set theory, which is currently being studied.
Significance of this book
A modern mathematical theory must start from a collection of axioms, i.e., an axiomatic system, and must be developed according to a correct logic.
Set theory is no exception.
There are several axiom systems in modern set theory
What can be logically derived from the representative ones?
The axioms of classical set theory.
If you want to study or research axiomatic set theory
It is not possible to write a correct knowledge of classical set theory
The difference between classical set theory and axiomatic set theory is
The difference between classical set theory and axiomatic set theory is the same as the difference between the geometry of Euclid's "Original Theory" and the geometry starting from Hilbert's "Foundations of Geometry".
The physical appearance is different, but not even the content is different.
Classical set theory alone is sufficient for the study and research of other areas of mathematics
Preface
Set theory is an important field lying at the deepest foundations of mathematics
Guide
This book is an introduction to the study of set theory, a branch of mathematics.
What is set theory?
I don't know how many there are, but there are some cups and some cup dishes.
One by one, the cups are placed on top of another cup and plate.
If at the end there are no more cups and no more plates, we conclude that there were the same number of cups and plates.
If there are two groups of some kind, and one group of cups is added to the other group of cups and plates, and there is no excess or deficiency
Both groups necessarily contain the same number of members.
Let's call this the "first principle".
A group that is part of some group has fewer members than the original group
The whole is greater than its parts.
Let's call this the "second principle.
Problems with the above two principles
The even numbers 2,4,6... is a natural number 1,2,3... is a part of
Part I. Algebra of Sets
I Concept of a set
1 What is a set?
2 Sets and Exons
3 Representations of sets
4 Subsets
II Operations on Sets
1 Difference of sets
2 The empty set
3 Complementary sets
4 Sum
5 Common part
6 Relationships between the nominal operations
7 Set families
III Functions and Direct Product
1 What is a function?
2 Concepts Surrounding Functions
3 Composite Functions
4 One-to-One Correspondence
5 Direct product
6 Graphing Functions
Part 2 Concentration
I Concept of Concentration
1 What is concentration?
2 Definition of concentration
3 Countable sets
4 Unnumbered sets
II Concentration
1 Compensation for concentration
2 Berstein's theorem
3 Concentration of a wide set
III Sum of concentrations
1 Definition and properties of sum of concentrations
2 Set system
3 Analysis of the concept of sum of concentrations
4 Extension of the sum of concentrations
IV Product of concentrations
1 Definition and properties of product of concentrations
2 Relationship between sum and product
3 Extension of the product of concentrations
V Width of concentration
1 Definition of width
2 Properties of width
Part 3 Ordered Numbers
I Order
1 Order
2 Ordered sets
3 Isomorphisms
4 ordinal type
II Alignment set
1 Aligned sets
2 Comparison of integer sets
III Ordered number
1 Permutation number
2 Order numbers, small and large
3 Sum of ordinal numbers
4 Product of ordinal numbers
5 Transfinite induction
6 Definition of the width of ordinal numbers
7 Properties of the width
8 Epsilon-numbers
9 Relationship between ordinal numbers and concentration
IV Alignability theorem
1 Selection axiom
2 The Orderability Theorem
3 Applications of the orderability theorem
Conclusion
Appendix
1 Sets and Logic
2 Zorn's Corollary
3 Axioms of Set Theory
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