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))
This says that for every property a(x) there is a set y that has all
the elements that satisfy a(x).
Notation: this y = {x∣a(x)}
Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift)
1879, 1903)
(∀z)(z ∈ {x∣a(x)}) ⇔ a(z)
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))
This says that for every property a(x) there is a set y that has all
the elements that satisfy a(x).
Notation: this y = {x∣a(x)}
Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift)
1879, 1903)
(∀z)(z ∈ {x∣a(x)}) ⇔ a(z)
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))
This says that for every property a(x) there is a set y that has all
the elements that satisfy a(x).
Notation: this y = {x∣a(x)}
Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift)
1879, 1903)
(∀z)(z ∈ {x∣a(x)}) ⇔ a(z)
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))
This says that for every property a(x) there is a set y that has all
the elements that satisfy a(x).
Notation: this y = {x∣a(x)}
Comprehension Axiom (Cantor 1874, Frege (Begriffsschrift)
1879, 1903)
(∀z)(z ∈ {x∣a(x)}) ⇔ a(z)
Examples of Sets
{x∣x = a} = {a}
{x∣x 2 = 1} = {1, −1}
{x∣x = 1 ∨ x = −1} = {1, −1}
{x∣x ∈ Z ∧ (∀y )(y ∈ Z ⇒ (∃z)(z ∈ Z ∧ y = zx)} = {1, −1}
⇑ x∣y
{x∣x ∈ R ∧ 0 < x < 1} = (0, 1)
Extensionality Axiom
A = B ⇔ (∀x)(x ∈ A ⇔ x ∈ B)
Sets are equal if they have the same elements.
Assume if a(x) is a property of x:
A = B ⇒ (a(A) ⇔ a(B))
Special Sets
{x∣x ≠ x} = {x∣x ∈ R ∧ x 2 = −1} = ∅
Empty Set
{x∣x = x} = V
Universal Set
{x∣x ∈ A ∨ x ∈ B} = A ∪ B
Union of A and B
{x∣x ∈ A ∧ x ∈ B} = A ∩ B Intersection of A and B
{x∣¬x ∈ A} = A′
Complement of A
Subsets
Definition
A is a subset of B
A ⊂ B ⇔ (∀x)(x ∈ A ⇒ x ∈ B)
{x∣x ⊂ A} = P(A)(the powerset of A)
{x∣x ⊂ ∅} = {∅}
{x∣x ⊂ {∅} = {∅, {∅}}
{x∣x ⊂ {a, b, c}} = { ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
The Properties of Sets Come from Logic
e.g. A ∪ B = B ∪ A
Proof
A ∪ B = {x∣x ∈ A ∨ x ∈ B}
B ∪ A = {x∣x ∈ B ∨ x ∈ A}
By the Comprehension Axiom:
x ∈A∪B ⇔x ∈A∨x ∈B
⇔ x ∈ B ∨ x ∈ A by logic
⇔x ∈B ∪A
So (∀x)(x ∈ A ∪ B ⇔ x ∈ B ∪ A).
By the Extensionality Axiom A ∪ B = B ∪ A.
Similarly . . .
A∩B =B ∩A
(A ∪ B) ∪ C = A ∪ (B ∪ C )
A ∩ (B ∩ C ) = (A ∩ B) ∩ C
A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )
A ∪ (B ∩ C ) = (A ∩ B) ∪ (A ∩ C )
(A ∪ B)′ = A′ ∩ B ′
(A ∩ B)′ = A′ ∪ B ′
Comparing Sets
People
People
People
Chickens
Chickens
–
–
–
–
–
Heads
Brains
Chairs
Beaks
Tails
Definition
Equivalent Sets are sets that have elements matched by a one to
one mapping.
A ≈ B ⇔ (∃f )f ∶ A → B ∧ f is 1-1 and onto (a bijection)
A is equivalent to B
A ⪯ B ⇔ (∃f )f ∶ A → B ∧ f is 1 -1
A ≺ B ⇔ A ⪯ B ∧ ¬A ≈ B
Examples
{+, −, ÷} ≈ {2, , ,}
{+, −, ÷} ≺ {2, , ,, /}
Defining Natural Numbers
0=∅
1 = {0} = {∅}
2 = {0, 1}
3 = {0, 1, 2} etc.
Note
0 ≺ 1 ≺ 2 ≺ 3 ≺ ...
{+, −, ÷} ≈ {0, 1, 2} = 3.
Definition
If A ≈ n, n is the cardinal number of A, or n = #A.
So, #{+, −, ÷} = 3.
Cardinal Arithmetic
Definition
Addition: If A ∩ B = ∅, #A + #B = #(A ∪ B).
e.g. 3 + 2 = #{0, 1, 2} + #{3, 4} = #{0, 1, 2, 3, 4} = 5.
Definition
Multiplication: #A ⋅ #B = #(A × B) where
A × B = {(a, b)∣a ∈ A, b ∈ B}.
e.g.
3 ⋅ 2 = #{0, 1, 2} ⋅ #{0, 1}
= #({0, 1, 2} × {0, 1})
= #{(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)}
=6
Infinite Cardinal Numbers
Let #{1, 2, 3, 4, . . . } = ℵ0 .
If N = {1, 2, 3, 4, . . . }, E = {2, 4, 6, 8, . . . }.
then if f (n) = 2n, f ∶ N → E , f is 1-1 and onto.
So N ≈ E and #E = ℵ0 .
Also Q = {0, −1, 1, − 21 , 21 , −2, 2, − 13 , 13 , − 32 , 23 , −3, 3, . . . }.
So N ≈ Q and #Q = ℵ0 .
Some Arithmetic
For n ∈ N:
ℵ0 + n = ℵ0
nℵ0 = ℵ0
ℵn0 = ℵ0
What about nℵ0 ?
Is N ≈ [0, 1) ≈ R?
nℵ0
Assume N ≈ [0, 1). Then there is an f ∶ N → [0, 1). f is 1-1 and
onto
f (1) = .a11 a12 . . .
So:
f (2) = .a21 a22 . . .
f (3) = .a31 a32 . . .
⋮
Let b = .b1 b2 b3 . . .
⎧
⎪
⎪aii + 1 if aii = 0
where bi = ⎨
⎪aii − 1 if aii = 0
⎪
⎩
Then b ≠ f (n) for any n
So N ≺ [0, 1) ≈ R
#[0, 1) = #R = 2ℵ0 .
These must all be different
(for 1-1) and use up all of
[0, 1) (for onto)
Cantor
#N = ℵ0 is the smallest infinite cardinal number.
ℵ1 is the next smallest,
then ℵ2 , etc.
Continuum
2ℵ0 = ℵ1
Hypothesis:
Generalised
Continuum
Hypothesis: 2ℵn = ℵn+1
Unresolved
1964.
Georg Cantor (1845 - 1918) 1874
problem
till
Cantor’s Theorem
A ≺ P(A)
(Like the proof of N ≺ [0, 1))
Cantor’s Paradox, 1899
V = {x∣x = x}
So
P ⊂V
So
P ⪯V
But
V ≺ P(V )
i.e.
¬P(V ) ⪯ V .
Burali-Forti Paradox, 1897 Involves Ordinal Numbers.
Russel and Frege
Bertrand Russell (1872 1970)
Gottlob Frege (1845 - 1925)
Russell sent his paradox to Frege when volume 2 of his
Begriffsschrift was about to be published.
Russel’s Paradox (1903)
If R = {x∣x ∈/ x}, then by the Comprehension axiom:
y ∈ {x∣a(x)} ⇔ a(y )
R ∈ R ⇔ R ∈/ R
A contradiction! (Gives: R ∈ R ∧ ¬R ∈ R)
Hilbert at the 1900 International Congress of mathematics
in Paris
What would be the major problems of interest in
mathematics in the 20th
century?
Hilbert listed 23:
1 The Continuum
hypothesis, is
2ℵ0 = ℵ1 ?
2 Prove that the
axioms of arithmetic
are consistent.
David Hilbert (1862 - 1943)
The Hilbert Programme
Aim: To ground all existing theories (including set theory) to a
finite, complete set of axioms, and prove that these are consistent.
Gödel’s Incompleteness Theorems 1931
1. In any axiomatic
systems, strong
enough to include
arithmetic, there is
always an arithmetical
statement that can
neither be proved nor
disproved.
Kurt Gödel (1906 - 1978)
2. A statement saying
such a system is
consistent can neither
be proved nor
disproved.
Ways of Beating the Paradoxes: Change the
Logic
Intuitionistic Logic
Luitzen Brouwer (1881 1966) - 1908
Arend Heyting (1898 - 1980)
- 1930
Don’t allow ⊢ P ∨ ¬P, so you don’t have ⊢ V ≺ V ∨ ¬V ≺ V or
⊢ R ∈ R ∨ ¬R ∈ R.
Paraconsistent Logic
Newton da Costa (1929 - !)
Allow ⊢ R ∈ R and ⊢ ¬R ∈ R, but not ⊢ P ∧ ¬P ⇒ Q.
(This doesn’t work, Curry’s Paradox, 1942 - IF C = {x∣x ∈ x ⇒ X }
then ⊢ X .)
Logic and Aritmetic based on Combinatory Logic
Moses Schonfinkel (1887 1942) - 1924
Haskell Curry (1900 - 1982) 1930
Logic and Arithmetic based on Lambda Calculus
Alonzo Church (1903 - 1995) - 1933
Ways of Beating the Paradoxes: Change the Set
Theory
Change the Set Theory
Ernst Zermelo (1871 - 1953)
- 1908
Abraham Fraenkel (1891 1965) - 1921
Axioms
(∀z)(z ∈ x ⇔ z ∈ y ) ⇒ x = y
Extensionality
(∃y )(∀x)(x ∈ y ⇔ x ∈ z ∧ P(x)) Substitution of Specification
(Defines subsets of z)
(∃z)(x ∈ z ∧ y ∈ z)
Pairing
(defines {x, y })
(∃z)(∀x)(x ∈ y ∧ y ∈ w ⇒ x ∈ z)
Union
(z is the set of the elements of the elements of w .)
+ Others
von Neumann, Bernays and Gödel
John von Neumann (1903 1957) - 1925, 1928
Paul Bernays (1888 - 1977) 1937, 1945
Gödel - 1940
Has sets and classes (only sets can be elements of classes.)
Axioms
(∀z)(z ∈ X ⇔ z ∈ Y ) ⇒ X = Y
(X, Y classes)
(∃Y )(x ∈ Y ⇔ a(x)) (Y is a class)
Extensionality
comprehension
Only y ∈ X if y is a set
+ Axioms for sets
Continuum Hypothesis
2ℵ0 = ℵ1
Can neither be proved or
disproved in NBG or ZF!
Paul Cohen (1934 - 2007)
Cardinal Numbers and Measures
The measure, or length, of:
[m, m + n] = n
#[m, m + n] = 2ℵ0
#[0, a] = 2ℵ0 if a > 0
IF #A = ℵ0 , the measure A = 0
1
2
3
, n+1 , n+2 , . . . }
n
10 10
10
k
can be made arbitrarily small for n large
10n+k−1
N≈{
Question:If #A = 2ℵ0 is a measure for A > 0?