Elements of Set Theory - CM0832
Andrés Sicard-Ramírez
Universidad EAFIT
Semester 2017-2
Administrative Information
Course web page
http://www1.eafit.edu.co/asr/courses/set-theory-CM0832/
Exams, bibliography, etc.
See course web page.
Elements of Set Theory - CM0832. Introduction
2/105
Notation
Logical constants
∧
∨
⇒
¬
⇔
⊥
∀𝑥
∃𝑥
(and)
(or)
(if , then )
(not)
(if and only if, iff)
(falsity)
(for every 𝑥)
(there exists a 𝑥)
conjunction
inclusive∗ disjunction
implication
negation
bi-implication, equivalence
bottom, falsum
universal quantifier
existential quantifier
Sets
Sets will be denote by lowercase letters (𝑎, 𝑏, …), uppercase letters
(𝐴, 𝐵, …), script letters (𝒜, ℬ, …) and Greek letters (𝛼, 𝛽, …).
∗
One or the other or both.
Elements of Set Theory - CM0832. Introduction
3/105
Origins
Georg Cantor (1845 – 1918)
Elements of Set Theory - CM0832. Introduction
4/105
Origins
“Set theory was invented by Georg Cantor…It was however Cantor who realized
the significance of one-to-one functions
between sets and introduced the notion
of cardinality of a set.” (p. 15)
Elements of Set Theory - CM0832. Introduction
5/105
Origins
“Set theory was born on that December 1873 day when Cantor established
that the reals are uncountable, i.e.
there is no one-to-one correspondence between the reals and the natural
numbers.” (p. XII)
Elements of Set Theory - CM0832. Introduction
6/105
Naive Set Theory
Remark
Cantor set theory is also called naive set theory.
Elements of Set Theory - CM0832. Introduction
7/105
Naive Set Theory
Remark
Cantor set theory is also called naive set theory.
Definition (Cantor’s set definition)
“Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres
Denkens (welche die Elemente von M genannt werden) zu einem Ganzen.” [Cantor 1895, p. 1]
Translation: “A set is a collection into a whole of definite, distinct objects
of our intuition or our thought. The objects are called elements (members)
of the set.” [Hrbacek and Jech 1999, p. 1]
Elements of Set Theory - CM0832. Introduction
8/105
Naive Set Theory
Membership relation (a binary relation)
𝑡 ∈ 𝐴 means that 𝑡 is a member of 𝐴.
The expression 𝑡 ∉ 𝐴 is an abbreviation of ¬(𝑡 ∈ 𝐴).
Elements of Set Theory - CM0832. Introduction
9/105
Naive Set Theory
Membership relation (a binary relation)
𝑡 ∈ 𝐴 means that 𝑡 is a member of 𝐴.
The expression 𝑡 ∉ 𝐴 is an abbreviation of ¬(𝑡 ∈ 𝐴).
Principle of extensionality, empty set, pair set, union and intersection,
subset and power set
See whiteboard.
Elements of Set Theory - CM0832. Introduction
10/105
Naive Set Theory
Issue: Implicit use of properties of sets
∗
†
Figure from https://commons.wikimedia.org/w/index.php?curid=48219447
See Moore [1982, p. 14] or Hrbacek and Jech [1999, pp. 145-6].
Elements of Set Theory - CM0832. Introduction
11/105
Naive Set Theory
Issue: Implicit use of properties of sets
Example (axiom of choice∗ )
Illustration of the axiom.
∗
†
Figure from https://commons.wikimedia.org/w/index.php?curid=48219447
See Moore [1982, p. 14] or Hrbacek and Jech [1999, pp. 145-6].
Elements of Set Theory - CM0832. Introduction
12/105
Naive Set Theory
Issue: Implicit use of properties of sets
Example (axiom of choice∗ )
Illustration of the axiom.
The proof that a real function 𝑓 is continuous at a point 𝑝 if and only if 𝑓
is sequentially continuous at 𝑝 [Heine 1872] requires of the axiom of
choice.†
∗
†
Figure from https://commons.wikimedia.org/w/index.php?curid=48219447
See Moore [1982, p. 14] or Hrbacek and Jech [1999, pp. 145-6].
Elements of Set Theory - CM0832. Introduction
13/105
Naive Set Theory
Issue: Too general method of abstraction
Elements of Set Theory - CM0832. Introduction
14/105
Naive Set Theory
Issue: Too general method of abstraction
Example (Russell’s paradox)
See whiteboard.
Elements of Set Theory - CM0832. Introduction
15/105
Russell’s Paradox
Gottlob Frege
Bertrand Russell
(1848 – 1925)
(1872 – 1970)
Elements of Set Theory - CM0832. Introduction
16/105
Letter from Russell to van Heijenoort∗
Penrhyndeudraeth, 23 November 1962
Dear Professor van Heijenoort,
As I think about acts of integrity and grace, I realise there is nothing in my
knowledge to compare with Frege’s dedication to truth. His entire life’s
work was on the verge of completion, much of his work had been ignored
to the benefit of men infinitely less capable, his second volume was about
to be published, and upon finding that his fundamental assumption was in
error, he responded with intellectual pleasure clearly submerging any
feelings of personal disappointment. It was almost superhuman and a
telling indication of that of which men are capable if their dedication is to
creative work and knowledge instead of cruder efforts to dominate and be
known.
Yours sincerely
Bertrand Russell
∗
[van Heijenoort 1967, p. 127].
Elements of Set Theory - CM0832. Introduction
17/105
Naive Set Theory
Issue: Too general method of abstraction
Exercise
Which is the paradox called Berry paradox?
Elements of Set Theory - CM0832. Introduction
18/105
None of our later work will actually depend on this informal description,
but we hope it will illuminate
Informally
Building Sets∗ the motivation behind some of the things
we will do.
IX
w
o
Fig.2. Vo is the set A of atoms.
First we gather together all those things that are not themselves sets but
we want to have as members
such things atoms. For
𝑉that
𝑉𝜔 =of𝑉0sets.
∪ 𝑉Call
0 = 𝐴 (set of atoms)
1 ∪ ⋯ 𝑉𝛼+1 = 𝑉𝛼 ∪ 𝒫𝑉𝛼
example, if we want to be able to speak of the set of all two-headed
𝑉𝑛+1
= 𝑉then
𝒫𝑉must
= 𝑉coins
coins,
include𝑉𝜔+1
all such
in 𝜔
our collection of atoms.
𝑛 ∪ we
𝑛
𝜔 ∪ 𝒫𝑉
∗
Let A be the set of all atoms; it is the first set in our description.
Fig. 2 We
[Enderton
1977]. to build up a hierarchy
now proceed
Elements of Set Theory - CM0832. Introduction
19/105
Our theory then will ignore all objects that are not sets (as interesting
and real as such objects may be).
Informally
Building Sets∗ Instead we will concentrate just on "pure"
sets that can be constructed without the use of such external objects. In
1
v.
1
v.,
I
Fig. 3. The ordinals are the backbone of the universe.
The ordinal numbers are the backbone of the universe.
particular, any member of one of our sets will itself be a set, and each of its
members, if any, will be a set, and so forth. (This does not produce an
infinite regress,
we stop when we reach 0.)
𝑉0 =because
∅ (no atoms)
𝑉𝛼+1 = 𝒫𝑉𝛼
Now that
we have banished atoms, the picture
becomes narrower
∗
(Fig.
3). The construction
is also simplified. We have defined ~+1 to be
Fig.
3 [Enderton
1977].
J<. uTheory
£!l>V-aCM0832.
. Now Introduction
it turns out that this is the same as A u £!l>Va (see Exercise
Elements of Set
20/105
Classes
Description
A set is a class, but some classes are too large to be a sets.
Example
The collection of all sets.
Remark
A class A is a set if 𝐴 ⊆ 𝑉𝛼 (𝐴 ∈ 𝑉𝛼+1 ) for some 𝛼.
Elements of Set Theory - CM0832. Introduction
21/105
Axiomatic Method
Some features
Axioms: Explicitly list of assumptions
Elements of Set Theory - CM0832. Introduction
22/105
Axiomatic Method
Some features
Axioms: Explicitly list of assumptions
Theorems: Logical consequence of the axioms
Elements of Set Theory - CM0832. Introduction
23/105
Axiomatic Method
Some features
Axioms: Explicitly list of assumptions
Theorems: Logical consequence of the axioms
Property of set theory: It should be an axiom or a theorem
Elements of Set Theory - CM0832. Introduction
24/105
Axiomatic Method
Axiomatic set theory as a fundational system for mathematics
“Mathematics can be embedded into set theory”
“…our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics—a remarkable fact.” [Enderton
1977, p. 11]
“Experience has shown that practically all notions used in contemporary mathematics can be defined, and their mathematical properties
derived, in this axiomatic system. In this sense, the axiomatic set
theory serves as a satisfactory foundations for the other branches of
mathematics.” [Hrbacek and Jech 1999, p. 3]
Elements of Set Theory - CM0832. Introduction
25/105
Axiomatic Method
Some axiomatic systems
Zermelo-Fraenkel set theory with choice (ZFC)
von Neumann–Bernays–Gödel set theory
Tarski–Grothendieck set theory
Elements of Set Theory - CM0832. Introduction
26/105
Axiomatic Method
First-Order Theories∗
“The adjective ‘first-order’ is used to distinguish the languages…from those
in which are predicates having other predicates or functions as arguments,
or quantification over functions or predicates, or both.” [Mendelson 1997,
p. 56]
∗
For an introduction to first-order languages and first-order theories see, for example,
[Hamilton 1978] or [Mendelson 1997].
Elements of Set Theory - CM0832. Introduction
27/105
Axiomatic Method
First-Order Theories∗
“The adjective ‘first-order’ is used to distinguish the languages…from those
in which are predicates having other predicates or functions as arguments,
or quantification over functions or predicates, or both.” [Mendelson 1997,
p. 56]
Primitive notions
We only need two primitive notions, the concepts of “set” and “member”.
∗
For an introduction to first-order languages and first-order theories see, for example,
[Hamilton 1978] or [Mendelson 1997].
Elements of Set Theory - CM0832. Introduction
28/105
Axioms and Operations
Extensionality Axiom
Extensionality axiom
If two sets have exactly the same members, then they are equal, that is,
∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].
Elements of Set Theory - CM0832. Axioms and Operations
30/105
Extensionality Axiom
Extensionality axiom
If two sets have exactly the same members, then they are equal, that is,
∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].
Remark
A set is pure if its members are also sets.
Elements of Set Theory - CM0832. Axioms and Operations
31/105
Extensionality Axiom
Extensionality axiom
If two sets have exactly the same members, then they are equal, that is,
∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].
Remark
A set is pure if its members are also sets.
Question
Have we any set? No, we haven’t.
Elements of Set Theory - CM0832. Axioms and Operations
32/105
Some Axioms for Building Sets
Empty (existence) axiom
There is a set having no members, that is,
∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).
Elements of Set Theory - CM0832. Axioms and Operations
33/105
Some Axioms for Building Sets
Empty (existence) axiom
There is a set having no members, that is,
∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).
Pairing axiom
For any sets 𝑢 and 𝑣, there is a set having as members just 𝑢 and 𝑣, that
is,
∀𝑎 ∀𝑏 ∃𝐶 ∀𝑥 (𝑥 ∈ 𝐶 ⇔ 𝑥 = 𝑎 ∨ 𝑥 = 𝑏).
Elements of Set Theory - CM0832. Axioms and Operations
34/105
Some Axioms for Building Sets
Union axiom (first version)
For any sets 𝑎 and 𝑏, there is a set whose members are those sets belonging
either to 𝑎 or to 𝑏 (or both), that is,
∀𝑎 ∀𝑏 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏).
Power set axiom
For any set 𝑎, there is a set whose members are exactly the subsets of 𝑎,
that is,
∀𝑎 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ⊆ 𝑎),
where
𝑢 ⊆ 𝑣 ⇔ ∀𝑡 (𝑡 ∈ 𝑢 ⇒ 𝑡 ∈ 𝑣).
Elements of Set Theory - CM0832. Axioms and Operations
35/105
Subset Axiom Scheme
Introduction
See whiteboard.
Elements of Set Theory - CM0832. Axioms and Operations
36/105
Subset Axiom Scheme
Introduction
See whiteboard.
Abstractions from the empty, pairing, union and power set axioms
Let 𝑎, 𝑏, 𝑢 and 𝑣 be sets, then
∅ = {𝑥 ∣ 𝑥 ≠ 𝑥},
{𝑢, 𝑣} = {𝑥 ∣ 𝑥 = 𝑢 ∨ 𝑥 = 𝑣},
𝑎 ∪ 𝑏 = {𝑥 ∣ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏},
𝒫𝑎 = {𝑥 ∣ 𝑥 ⊆ 𝑎}.
Elements of Set Theory - CM0832. Axioms and Operations
37/105
Subset Axiom Scheme
Subset axiom scheme (axiom scheme of comprehension/separation)
For each formula 𝜑, not containing 𝐵, the following is an axiom:
∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).
Elements of Set Theory - CM0832. Axioms and Operations
38/105
Subset Axiom Scheme
Subset axiom scheme (axiom scheme of comprehension/separation)
For each formula 𝜑, not containing 𝐵, the following is an axiom:
∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).
Remark
We stated an axiom scheme!
Elements of Set Theory - CM0832. Axioms and Operations
39/105
Subset Axiom Scheme
Subset axiom scheme (axiom scheme of comprehension/separation)
For each formula 𝜑, not containing 𝐵, the following is an axiom:
∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).
Remark
We stated an axiom scheme!
Abstraction from the subset axiom scheme
{𝑥 ∈ 𝑐 ∣ 𝜑} is the set of all 𝑥 ∈ 𝑐 satisfying the property 𝜑.
Elements of Set Theory - CM0832. Axioms and Operations
40/105
Subset Axiom Scheme
Theorem (Enderton [1977], Theorem 2A)
There is no set to which every set belongs.
Elements of Set Theory - CM0832. Axioms and Operations
41/105
Subset Axiom Scheme
Theorem (Enderton [1977], Theorem 2A)
There is no set to which every set belongs.
Exercise
Why does the subset axiom scheme avoid the Berry paradox?
Elements of Set Theory - CM0832. Axioms and Operations
42/105
Arbitrary Unions and Intersections
Definition
Let 𝐴 be a set, then
⋃ 𝐴 = {𝑥 ∣ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏}.
Examples
𝑎 ∪ 𝑏 = ⋃{𝑎, 𝑏},
⋃{𝑎} = 𝑎,
⋃ ∅ = ∅.
Elements of Set Theory - CM0832. Axioms and Operations
43/105
Arbitrary Unions and Intersections
Definition
Let 𝐴 be a set, then
⋃ 𝐴 = {𝑥 ∣ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏}.
Examples
𝑎 ∪ 𝑏 = ⋃{𝑎, 𝑏},
⋃{𝑎} = 𝑎,
⋃ ∅ = ∅.
Union axiom (final version)
For any set 𝐴, there exists a set 𝐵 whose elements are exactly the members
of the members of 𝐴, that is,
∀𝐴 ∃𝐵 ∀𝑥 [𝑥 ∈ 𝐵 ⇔ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏)].
Elements of Set Theory - CM0832. Axioms and Operations
44/105
Arbitrary Unions and Intersections
Definition
Let 𝐴 be a set, then
⋂ 𝐴 = {𝑥 ∣ ∀𝑏 (𝑏 ∈ 𝐴 ⇒ 𝑥 ∈ 𝑏}, for 𝐴 ≠ ∅.
Elements of Set Theory - CM0832. Axioms and Operations
45/105
Arbitrary Unions and Intersections
Definition
Let 𝐴 be a set, then
⋂ 𝐴 = {𝑥 ∣ ∀𝑏 (𝑏 ∈ 𝐴 ⇒ 𝑥 ∈ 𝑏}, for 𝐴 ≠ ∅.
Theorem (Enderton [1977], Theorem 2B)
For any nonempty set 𝐴, there exists a unique set 𝐵 such that for any 𝑥,
𝑥 ∈ 𝐵 ⇔ 𝑥 belongs to every member of 𝐴.
Elements of Set Theory - CM0832. Axioms and Operations
46/105
Algebra of Sets
Exercise (Enderton [1977], Exercise 2.18)
Assume that 𝐴 and 𝐵 are subsets of 𝑆. List all of the different sets that
can be made from these three by use of the binary operations ∪, ∩, and −.
Elements of Set Theory - CM0832. Axioms and Operations
47/105
Algebra of Sets
Exercise (Enderton [1977], Exercise 2.18)
Assume that 𝐴 and 𝐵 are subsets of 𝑆. List all of the different sets that
can be made from these three by use of the binary operations ∪, ∩, and −.
The Venn diagram shows four possible regions for shading, that is, we
have 24 different sets given by
∅, 𝐴, 𝐵, 𝑆, 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, 𝐴 − 𝐵, 𝐵 − 𝐴, 𝐴 + 𝐵, 𝑆 − 𝐴, 𝑆 − 𝐵,
𝑆 − (𝐴 ∪ 𝐵), 𝑆 − (𝐴 ∩ 𝐵), 𝑆 − (𝐴 − 𝐵), 𝑆 − (𝐵 − 𝐴) and 𝑆 − (𝐴 + 𝐵),
where the binary operation + is the symmetric difference defined by
𝐴 + 𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)
= (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵).
Elements of Set Theory - CM0832. Axioms and Operations
48/105
Relations and Functions
Ordered Pairs
Exercise
Let 𝑎 and 𝑏 be sets. The order pair ⟨𝑎, 𝑏⟩ should be a set such that
⟨𝑎, 𝑏⟩ = ⟨𝑐, 𝑑⟩ ⇔ 𝑎 = 𝑐 ∧ 𝑏 = 𝑑.
We can define the order pair ⟨𝑎, 𝑏⟩ via Kuratowski’s definition
⟨𝑎, 𝑏⟩ = {{𝑥}, {𝑥, 𝑦}}.
To give a different definition of order pairs.
Elements of Set Theory - CM0832. Relations and Functions
50/105
Relations
Definition
A relation is a set of ordered pairs.
Elements of Set Theory - CM0832. Relations and Functions
51/105
Relations
Definition
A relation is a set of ordered pairs.
Definition
We define the domain, range and field of 𝑅 by
𝑥 ∈ dom 𝑅 ⇔ ∃𝑦 ⟨𝑥, 𝑦⟩ ∈ 𝑅,
𝑥 ∈ ran 𝑅 ⇔ ∃𝑡 ⟨𝑡, 𝑥⟩ ∈ 𝑅,
fld 𝑅 = dom 𝑅 ∪ ran 𝑅.
Elements of Set Theory - CM0832. Relations and Functions
52/105
Functions
Definition
A function is a relation 𝐹 such that for each 𝑥 in dom 𝐹 there is only one 𝑦
such that 𝑥𝐹 𝑦.
Elements of Set Theory - CM0832. Relations and Functions
53/105
Functions
Definition
A function is a relation 𝐹 such that for each 𝑥 in dom 𝐹 there is only one 𝑦
such that 𝑥𝐹 𝑦.
Exercise (Enderton [1977], Exercise 3.11)
Prove the following version (for functions) of the extensionality principle:
Assume that 𝐹 and 𝐺 are functions, dom 𝐹 = dom 𝐺, and 𝐹 (𝑥) = 𝐺(𝑥)
for all 𝑥 in the common domain. Then 𝐹 = 𝐺.
Elements of Set Theory - CM0832. Relations and Functions
54/105
Functions
Definitions
Let 𝐴, 𝐹 and 𝐺 be sets.
𝐹 −1 = {⟨𝑢, 𝑣⟩ ∣ 𝑣𝐹 𝑢}
𝐹 ∘ 𝐺 = {⟨𝑢, 𝑣⟩ ∣ ∃𝑡 (𝑢𝐺𝑡 ∧ 𝑡𝐹 𝑣)}
(inverse of 𝐹 )
(composition of 𝐹 and 𝐺)
𝐹 ↾ 𝐴 = {⟨𝑢, 𝑣⟩ ∣ 𝑢𝐹 𝑣 ∧ 𝑢 ∈ 𝐴}
(restriction of 𝐹 to 𝐴)
𝐹 ⟦𝐴⟧ = ran (𝐹 ↾ 𝐴)
(image of 𝐴 under 𝐹 )
= {𝑣 ∣ (∃𝑢 ∈ 𝐴)𝑢𝐹 𝑣}
Elements of Set Theory - CM0832. Relations and Functions
55/105
Axiom of Choice
Axiom of choice (first form)
For any relation 𝑅 there is a function 𝐻 ⊆ 𝑅 with dom 𝐻 = dom 𝑅.
Elements of Set Theory - CM0832. Relations and Functions
56/105
Axiom of Choice
Axiom of choice (first form)
For any relation 𝑅 there is a function 𝐻 ⊆ 𝑅 with dom 𝐻 = dom 𝑅.
Remark
Is the axiom of choice accepted in constructive mathematics? (See, for
example, Martin-Löf [2006]).
Elements of Set Theory - CM0832. Relations and Functions
57/105
numbers.
Suppose Relations
we now define a binary relation R on A as follows: For
Equivalence
x and y in A,
Idea∗
xRy
¢>
x and yare in the same little box.
(a)
(b)
•
•
•
•
•
•
(c)
Fig. 12. Partitioning a set into six little boxes.
Then we can easily see that R has the following three properties.
1.
2.
yRx.
R is reflexive on A, by which we mean that xRx for all x E A.
R is symmetric, by which we mean that whenever xRy, then also
3. R is transitive, by which we mean that whenever xRy and yRz,
then also xRz.
Definition R is an equivalence relation on A iff R is a binary relation
on A that is reflexive on A, symmetric, and transitive.
∗
Theorem 3M If R is a symmetric and transitive relation, then R is an
Fig.
12 [Enderton
1977].
equivalence
relation
on fld R.
Elements of Set Theory - CM0832. Relations and Functions
58/105
numbers.
Suppose Relations
we now define a binary relation R on A as follows: For
Equivalence
x and y in A,
Idea∗
xRy
¢>
x and yare in the same little box.
(a)
(b)
•
•
•
•
•
•
(c)
Fig. 12. Partitioning a set into six little boxes.
Definitions
Let 𝑅Then
be awebinary
relation
onRahas
setthe
𝐴.following
The relation
𝑅 is
can easily
see that
three properties.
1. R is
on all
A, by
reflexive
iffreflexive
𝑥𝑅𝑥 for
𝑥 ∈which
𝐴, we mean that xRx for all x E A.
2.
yRx.
R is symmetric, by which we mean that whenever xRy, then also
3. R is transitive, by which we mean that whenever xRy and yRz,
then also xRz.
Definition R is an equivalence relation on A iff R is a binary relation
on A that is reflexive on A, symmetric, and transitive.
∗
Theorem 3M If R is a symmetric and transitive relation, then R is an
Fig.
12 [Enderton
1977].
equivalence
relation
on fld R.
Elements of Set Theory - CM0832. Relations and Functions
59/105
numbers.
Suppose Relations
we now define a binary relation R on A as follows: For
Equivalence
x and y in A,
Idea∗
xRy
¢>
x and yare in the same little box.
(a)
(b)
•
•
•
•
•
•
(c)
Fig. 12. Partitioning a set into six little boxes.
Definitions
Let 𝑅Then
be awebinary
relation
onRahas
setthe
𝐴.following
The relation
𝑅 is
can easily
see that
three properties.
1. R is
on all
A, by
reflexive
iffreflexive
𝑥𝑅𝑥 for
𝑥 ∈which
𝐴, we mean that xRx for all x E A.
2.
R is symmetric, by which we mean that whenever xRy, then also
symmetric
iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 and
yRx.
3. R is transitive, by which we mean that whenever xRy and yRz,
then also xRz.
Definition R is an equivalence relation on A iff R is a binary relation
on A that is reflexive on A, symmetric, and transitive.
∗
Theorem 3M If R is a symmetric and transitive relation, then R is an
Fig.
12 [Enderton
1977].
equivalence
relation
on fld R.
Elements of Set Theory - CM0832. Relations and Functions
60/105
numbers.
Suppose Relations
we now define a binary relation R on A as follows: For
Equivalence
x and y in A,
Idea∗
xRy
¢>
x and yare in the same little box.
(a)
(b)
•
•
•
•
•
•
(c)
Fig. 12. Partitioning a set into six little boxes.
Definitions
Let 𝑅Then
be awebinary
relation
onRahas
setthe
𝐴.following
The relation
𝑅 is
can easily
see that
three properties.
1. R is
on all
A, by
reflexive
iffreflexive
𝑥𝑅𝑥 for
𝑥 ∈which
𝐴, we mean that xRx for all x E A.
2.
R is symmetric, by which we mean that whenever xRy, then also
symmetric
iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 and
yRx.
3. R isifftransitive,
which
mean
transitive
(𝑥𝑅𝑦 ∧ by
𝑦𝑅𝑧)
⇒ we
𝑥𝑅𝑧
for that
all 𝑥,whenever
𝑦, 𝑧 ∈ 𝐴.xRy and yRz,
then also xRz.
Definition R is an equivalence relation on A iff R is a binary relation
on A that is reflexive on A, symmetric, and transitive.
∗
Theorem 3M If R is a symmetric and transitive relation, then R is an
Fig.
12 [Enderton
1977].
equivalence
relation
on fld R.
Elements of Set Theory - CM0832. Relations and Functions
61/105
Equivalence Relations
Questions
Let 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalence
relation?
Elements of Set Theory - CM0832. Relations and Functions
62/105
Equivalence Relations
Questions
Let 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalence
relation?
Let 𝐴 ≠ ∅ be a set. Are the relations ∅ and 𝐴 × 𝐴 equivalence
relations?
Elements of Set Theory - CM0832. Relations and Functions
63/105
Equivalence Relations
Questions
Let 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalence
relation?
Let 𝐴 ≠ ∅ be a set. Are the relations ∅ and 𝐴 × 𝐴 equivalence
relations?
Let 𝐴 be a singleton. It is possible to define an equivalence relation
on 𝐴?
Elements of Set Theory - CM0832. Relations and Functions
64/105
Linear Ordering Relations
Motivation
What means that 𝑅 is an ordering relation on a set 𝐴?
Elements of Set Theory - CM0832. Relations and Functions
65/105
Linear Ordering Relations
Motivation
What means that 𝑅 is an ordering relation on a set 𝐴?
Definition
Let 𝑅 be a binary relation on a set 𝐴. The relation 𝑅 satisfies trichotomy
if exactly one of the three alternatives
𝑥𝑅𝑦,
𝑥=𝑦
or 𝑦𝑅𝑥
holds for all for all 𝑥, 𝑦 ∈ 𝐴.
Elements of Set Theory - CM0832. Relations and Functions
66/105
Linear Ordering Relations
Definition
Let 𝐴 be a set. A linear (total) ordering on 𝐴 is a binary relation 𝑅 on 𝐴
such that:
1
𝑅 is transitive relation and
2
𝑅 satisfies trichotomy on 𝐴.
Elements of Set Theory - CM0832. Relations and Functions
67/105
Linear Ordering Relations
Definition
Let 𝐴 be a set. A linear (total) ordering on 𝐴 is a binary relation 𝑅 on 𝐴
such that:
1
𝑅 is transitive relation and
2
𝑅 satisfies trichotomy on 𝐴.
Example
⋮
3
2
1
0
Elements of Set Theory - CM0832. Relations and Functions
68/105
Natural Numbers
Natural Numbers
von Neumann’s construction
0 = ∅,
1 = {0} = {∅},
2 = {0, 1} = {∅, {∅}},
3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}},
⋮
Elements of Set Theory - CM0832. Natural Numbers
70/105
Natural Numbers
von Neumann’s construction
0 = ∅,
1 = {0} = {∅},
2 = {0, 1} = {∅, {∅}},
3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}},
⋮
Some properties
0 ∈ 1 ∈ 2 ∈ 3 ∈ ...
0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ ...
Elements of Set Theory - CM0832. Natural Numbers
71/105
Natural Numbers
A wrong impredicative definition
𝑛 = {0, 1, … , 𝑛 − 1}.
“We cannot just say that a set 𝑛 is a natural number if its elements are all
the smaller natural numbers, because such a “definition” would involve the
very concept being defined.”
[Hrbacek and Jech 1999, p. 40]
Elements of Set Theory - CM0832. Natural Numbers
72/105
Natural Numbers
Definition
Let 𝑎 be a set. The successor of 𝑎 is
𝑎+ = 𝑎 ∪ {𝑎}.
Elements of Set Theory - CM0832. Natural Numbers
73/105
Natural Numbers
Definition
Let 𝑎 be a set. The successor of 𝑎 is
Definition
𝑎+ = 𝑎 ∪ {𝑎}.
A set 𝐴 is called inductive iff
∅ ∈ 𝐴 and
if 𝑎 ∈ 𝐴 then 𝑎+ ∈ 𝐴.
Elements of Set Theory - CM0832. Natural Numbers
74/105
Natural Numbers
Definition
Let 𝑎 be a set. The successor of 𝑎 is
Definition
𝑎+ = 𝑎 ∪ {𝑎}.
A set 𝐴 is called inductive iff
∅ ∈ 𝐴 and
if 𝑎 ∈ 𝐴 then 𝑎+ ∈ 𝐴.
Infinity axiom
There exists an inductive set, that is,
(∃𝐴)[∅ ∈ 𝐴 ∧ ∀𝑎(𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].
Elements of Set Theory - CM0832. Natural Numbers
75/105
Natural Numbers
Definition
A natural number is a set that belongs to every inductive set.
Elements of Set Theory - CM0832. Natural Numbers
76/105
Natural Numbers
Definition
A natural number is a set that belongs to every inductive set.
Definition
The set of all natural numbers, denoted by 𝜔, is defined by
𝑥 ∈ 𝜔 ⇔ 𝑥 is a natural number.
Elements of Set Theory - CM0832. Natural Numbers
77/105
Induction Principle for the Natural Numbers
Induction principle for 𝜔 (one version)
Any inductive subset of 𝜔 coincides with 𝜔 [Enderton 1977, p. 69].
Elements of Set Theory - CM0832. Natural Numbers
78/105
Induction Principle for the Natural Numbers
Induction principle for 𝜔 (one version)
Any inductive subset of 𝜔 coincides with 𝜔 [Enderton 1977, p. 69].
Induction principle for 𝜔 (other version)
Let 𝑃 (𝑥) be a property. Assume that
i)
𝑃 (0) holds
ii)
For all 𝑛 ∈ 𝜔, 𝑃 (𝑛) implies 𝑃 (𝑛+ )
Then 𝑃 holds for all natural numbers 𝑛.
Proof.
“This is an immediate consequence of our definition of 𝑤. The assumptions
i) and ii) simple say that the set 𝐴 = {𝑛 ∈ 𝜔 ∣ 𝑃 (𝑛)} is inductive. 𝜔 ⊆ 𝐴
follows.” [Hrbacek and Jech 1999, p. 42]
Elements of Set Theory - CM0832. Natural Numbers
79/105
Recursion on the Natural Numbers
Theorem (Recursion on 𝜔 [Enderton 1977, p. 73])
Let 𝐴 be a set, 𝑎 ∈ 𝐴 and 𝐹 ∶ 𝐴 → 𝐴. Then there exists a unique function
ℎ ∶ 𝜔 → 𝐴 such that
ℎ(0) = 𝑎,
and for every 𝑛 in 𝜔
ℎ(𝑛+ ) = 𝐹 (ℎ(𝑛)).
Elements of Set Theory - CM0832. Natural Numbers
80/105
Induction as Foundations
“Thus inductive definibility is a
notion intermediate in strength
between predicate and fully
impredicative definability.
It would be interesting to formulate a
coherent conceptual framework that
made induction the principal notion.
There are suggestions of this in the
literature, but the possibility has not
yet been fully explored.” [Aczel 1977,
p. 780]
Elements of Set Theory - CM0832. Natural Numbers
81/105
Cardinal Numbers
Equinumerosity
Definition
A set 𝐴 is equinumerous to a set 𝐵, denoted 𝐴 ≈ 𝐵, iff there is a one-to-one
correspondence between 𝐴 and 𝐵.
Elements of Set Theory - CM0832. Cardinal Numbers
83/105
Equinumerosity
“The possibility that whole and part
may have the same number of terms
is, it must be confessed, shocking to
common-sense.” [Russell 1903,
p. 358]
Elements of Set Theory - CM0832. Cardinal Numbers
84/105
Equinumerosity
Theorem (Enderton [1977], Theorem 6B(a))
The set 𝜔 is not equinumerous to the set ℝ of real numbers.
Proof.
Let’s suppose 𝜔 ≈ ℝ, that is, there is an one-to-one correspondence
𝑓 ∶ 𝜔 → ℝ such that∗
𝑓(0) = 236.001 … ,
𝑓(1) = −7.777 … ,
𝑓(2) = 3.1415 … ,
⋮
(continues in the next slide)
∗
“We assume that a decimal expansion does not contain only the digit 9 from some
place on, so each real number has a unique decimal expansion.” [Hrbacek and Jech
1999, Theorem 6.1, p. 90]
Elements of Set Theory - CM0832. Cardinal Numbers
85/105
Equinumerosity
Proof (cont).
Let 𝑥 = 0.𝑑1 𝑑2 𝑑3 … ∈ ℝ, where
𝑑𝑛+1 = {
4
5
iff 𝑓(𝑛) ≠ 4,
iff 𝑓(𝑛) = 4.
The number 𝑥 doesn’t belong to the above enumeration. Therefore, ℝ is
non-enumerable.
Elements of Set Theory - CM0832. Cardinal Numbers
86/105
On Refutations of Cantor’s Diagonal Argument
“I dedicate this essay to the
two-dozen-odd people whose
refutations of Cantor’s diagonal
argument have come to me either as
referee or as editor in the last twenty
years or so... A few years ago it
occurred to me to wonder why so
many people devote so much energy
to refuting this harmless little
argument—what had it done to
make them angry with it?... These
pages report the results.” [Hodges
1998, p. 1]
Elements of Set Theory - CM0832. Cardinal Numbers
87/105
Finite Sets
Definition
A set is finite iff it is equinumerous to some natural number. Otherwise it
is infinite.
Elements of Set Theory - CM0832. Cardinal Numbers
88/105
The Continuum Hypothesis
The continuum hypothesis (CH)
There is no a set whose cardinality is strictly between the cardinality of the
set of the natural numbers and the cardinality of the set of real numbers,
that is,
2ℵ0 = ℵ1 .
CH could not be disproved [Gödel 1939] nor proved [Cohen 1963] in ZFC,
that is, CH is independent of ZFC set theory.
Elements of Set Theory - CM0832. Cardinal Numbers
89/105
Ordinal Numbers
Let’s Count
“Let us count—0, 1, 2, 3, 4, … . The natural or counting numbers start
with 0, the empty set, and continue with
1 = 0+ ,
2 = 1+ ,
3 = 2+ ,
etc. It turns out that is quite useful to continue the counting procedure
beyond natural numbers. Therefore, start with 𝜔 and define
𝜔 + 1 = 𝜔+ ,
𝜔 + 2 = (𝜔 + 1)+ ,
𝜔 + 3 = (𝜔 + 2)+ ,
etc.” (continues in the next slide)
Elements of Set Theory - CM0832. Ordinal Numbers
91/105
Let’s Count
(cont.)
“Let
𝑥 = {𝜔 + 𝑛 ∣ 𝑛 ∈ 𝜔}.
…Define,
𝑤 · 2 = 𝜔 + 𝜔 = 𝑥 ∪ 𝜔 = ⋃ 𝑥.
Now that we have extended the natural numbers sowewhat, it is easy to go
on to 𝑤 · 2 + 1, 𝑤 · 2 + 2, …, 𝜔 · 3, …, 𝜔 · 4, …, 𝜔 · 𝜔 = 𝜔2 ; and then start
over again with 𝜔2 + 1, 𝜔2 + 2, …, 𝜔2 + 𝜔, 𝜔2 + 𝜔 + 1, 𝜔2 + 𝜔 + 2, …,
𝜔2 + 𝜔 · 2, …, 𝜔2 + 𝜔 · 3, …, 𝜔2 · 2, …, 𝜔2 · 3, …, 𝜔3 , …, 𝜔4 , …, 𝑤𝜔 , etc.”
[Rubin 1967, p. 175]
Elements of Set Theory - CM0832. Ordinal Numbers
92/105
A Definition
Definition
The class of ordinal numbers, denoted by On, is defined by [Rubin 1967,
pp. 175-176] (see also [Enderton 1977, Corollary 7N])
0 ∈ On,
𝛼 ∈ On ⇒ 𝛼+ ∈ On,
𝐴 ⊆ On ⇒ ⋃ 𝐴 ∈ On.
Elements of Set Theory - CM0832. Ordinal Numbers
93/105
Ordinal Addition
Definition
Let 𝛼 and 𝛽 be ordinals. We define their addition by transfinite recursion
on 𝛽:
𝛼 + 0 = 𝛼,
𝛼 + 𝛽 + = (𝛼 + 𝛽)+ ,
𝛼 + 𝛽 = sup{𝛼 + 𝜆 ∣ 𝜆 < 𝛽}
= ⋃ (𝛼 + 𝜆), for all limit ordinal 𝛽.
𝜆<𝛽
Elements of Set Theory - CM0832. Ordinal Numbers
94/105
Ordinal Addition
Examples
𝜔 = 1 + 𝜔, 1 + 𝜔 ≠ 𝜔 + 1 and 𝜔 + 𝜔 ≠ 𝜔 + 1.
⋮
⟨2, 1⟩
⟨1, 1⟩
⋮
⋮
⟨0, 1⟩
3
⟨2, 1⟩
2
⟨1, 1⟩
⟨2, 0⟩
⟨2, 0⟩
1
⟨0, 1⟩
⟨1, 0⟩
⟨1, 0⟩
0
⟨0, 0⟩
⟨0, 0⟩
⟨0, 0⟩
(a) 𝜔
(b) 1 + 𝜔
(c) 𝜔 + 1
(d) 𝜔 + 𝜔
Elements of Set Theory - CM0832. Ordinal Numbers
⋮
⟨0, 1⟩
⋮
95/105
List of Axioms
Extensionality axiom: If two sets have exactly the same members,
then they are equal, that is,
∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].
Empty (existence) axiom: There is a set having no members, that is,
∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).
Pairing axiom: For any sets 𝑢 and 𝑣, there is a set having as members
just 𝑢 and 𝑣, that is,
∀𝑎 ∀𝑏 ∃𝐶 ∀𝑥 (𝑥 ∈ 𝐶 ⇔ 𝑥 = 𝑎 ∨ 𝑥 = 𝑏).
Elements of Set Theory - CM0832. List of Axioms
96/105
List of Axioms
Union axiom (first version): For any sets 𝑎 and 𝑏, there is a set whose
members are those sets belonging either to 𝑎 or to 𝑏 (or both), that is,
∀𝑎 ∀𝑏 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏).
Union axiom (final version): For any set 𝐴, there exists a set 𝐵 whose
elements are exactly the members of the members of 𝐴, that is,
∀𝐴 ∃𝐵 ∀𝑥 [𝑥 ∈ 𝐵 ⇔ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏)].
Power set axiom: For any set 𝑎, there is a set whose members are
exactly the subsets of 𝑎, that is,
∀𝑎 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ⊆ 𝑎).
Elements of Set Theory - CM0832. List of Axioms
97/105
List of Axioms
Subset axiom scheme (axiom scheme of comprehension or separation):
For each formula 𝜑, not containing 𝐵, the following is an axiom:
∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).
Infinity axiom: There exists an inductive set, that is,
∃𝐴 [∅ ∈ 𝐴 ∧ ∀𝑎 (𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].
Axiom of choice (a version): For any relation 𝑅 there is a function
𝐹 ⊆ 𝑅 with dom 𝐹 = dom 𝑅.
Elements of Set Theory - CM0832. List of Axioms
98/105
List of Axioms
Regularity (foundation) axiom: All sets are well-founded, that is,
∀𝐴 [𝐴 ≠ ∅ ⇒ ∃𝑚 (𝑚 ∈ 𝐴 ∧ 𝑚 ∩ 𝐴 = ∅)].
Elements of Set Theory - CM0832. List of Axioms
99/105
Appendix: Natural Deduction
Derivations Rules
𝜑
𝜓
∧I
𝜑∧𝜓
𝜑∧𝜓
∧E
𝜓
𝜑∧𝜓
∧E
𝜑
[𝜑]
𝜑
⋮
𝜓
→I
𝜑→𝜓
𝜑
∨I
𝜑∨𝜓
𝜓
∨I
𝜑∨𝜓
𝜑→𝜓
→E
𝜓
[𝜑]
[𝜓]
⋮
𝜎
⋮
𝜎
𝜑∨𝜓
𝜎
∨E
[¬𝜑]
⊥
𝜑 ⊥E
Elements of Set Theory - CM0832. Appendix: Natural Deduction
⋮
⊥
𝜑 RAA
101/105
Derivations Rules
Remarks
Abbreviations: ¬𝜑 ∶= 𝜑 → ⊥.
In the application of the →I rule, we may discharge zero, one, or more
occurrences of the assumption.
References: See, for example, [van Dalen 2013].
Elements of Set Theory - CM0832. Appendix: Natural Deduction
102/105
Examples
Example (van Dalen [2013], example p. 49)
Proof of ⊢ 𝜑 ∨ ¬𝜑.
[𝜑]1
𝜑 ∨ ¬𝜑
∨I
[¬(𝜑 ∨ ¬𝜑)]2
⊥
→I1
¬𝜑
∨I
𝜑 ∨ ¬𝜑
→E
[¬(𝜑 ∨ ¬𝜑)]2
⊥
𝜑 ∨ ¬𝜑
Elements of Set Theory - CM0832. Appendix: Natural Deduction
→E
RAA2
103/105
References
Aczel, Peter (1977). An Introduction to Inductive Definitions. In: Handbook of
Mathematical Logic. Ed. by Barwise, Jon. Vol. 90. Studies in Logic and the
Foundations of Mathematics. Elsevier. Chap. C.7.
Cantor, Georg (1895). Beiträge zur Begründung der transfiniten Mengenlehre.
Mathematische Annalen 46, pp. 481–512.
Cohen, Paul J. (1963). The Independence of the Continuum Hypothesis.
Proceedings of the National Academy of Sciences of the United States of
America 50.6, pp. 1143–1148.
Enderton, Herbert B. (1977). Elements of Set Theory. Academic Press.
Gödel, Kurt (1939). The Consistency of the Generalized Continuum-Hypothesis.
Bulletin of the American Mathematical Society 45.1, p. 93.
Hamilton, A. G. (1978). Logic for Mathematicians. Revised edition. Cambridge
University Press.
Hodges, Wilfrid (1998). An Editor Recalls some Hopeless Papers. The Bulletin of
Symbolic Logic 4.1, pp. 1–16.
Hrbacek, Karel and Jech, Thomas (1999). Introduction to Set Theory. Third
Edition, Revised and Expanded. Marcel Dekker.
Elements of Set Theory - CM0832. References
104/105
References
Martin-Löf, Per (2006). 100 Years of Zermelo’s Axiom of Choice: What was the
Problem with It? The Computer Journal 49.3, pp. 345–350.
Mendelson, Elliott (1997). Introduction to Mathematical Logic. 4th ed. Chapman
& Hall.
Moore, Gregory H. (1982). Zermelo’s Axiom of Choice. Its Origins, Development,
and Influence. Springer-Verlag.
Rubin, Jean E. (1967). Set Theory for the Mathematician. Holden-Day.
Russell, Bertrand (1903). The Principles of Mathematics. W. W. Norton &
Company, Inc.
van Dalen, Dirk (2013). Logic and Structure. 5th ed. Springer.
van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book in
Mathematical Logic, 1879-1931. Harvard University Press.
Elements of Set Theory - CM0832. References
105/105