a brief introduction to set theory. how many things?
TRANSCRIPT
A Brief Introduction to Set Theory
How Many Things?
Mereology
• Theory of parts and wholes• Are there bigger things than particles?• Arbitrary fusions• Nihilism?
How Many Things?
Lots of Little Things…
Some Weird Things
One Maximal Thing
Set Theory
• Sets are mathematical posits• Any time you have some things, there is a set containing those things• The set is a different thing• The things it contains are its members (not parts)• Since sets are things, they can be collected into sets
How Many Things?
How Many Things?
How Many Things?
How Many Things?
How Many Things?
How Many Things?
How Many Things?
History of Set Theory
• Founded by Georg Cantor in 1847.• Popular ever since.
Names for Sets
Extensive Notation
Bracket symbols
Extensive Notation
Names of the set’s members
John , Paul , George , Ringo
Extensive Notation
Order doesn’t matter
Paul , Ringo , John , George
Extensive Notation
Any name will do
Paul , Ringo , Dr. Winston O’Boogie , George
Intensive Notation
Variable (your choice: x, y, z, etc.)
x
Intensive Notation
Up and down line
x
Intensive Notation
Condition that uniquely picks out the set’s members
x x is a member of The Beatles
Intensive Notation
Condition that uniquely picks out the set’s members
x x had the most #1 British albums
Intensive Notation
Condition that uniquely picks out the set’s members
x x sang a song on Rubber Soul
Intensive Notation
Condition that uniquely picks out the set’s members
x x = John or x = Paul or x = George or x= Ringo
Set Membership
The fundamental relation in set theory is the set membership relation.
We write this relation with a stylized Greek epsilon: ϵ
Example: John ϵ { x | x is a member of The Beatles }
Set Membership
Only sets have members in this sense of “members”.
~(Ǝx)(x ϵ John)
Sets can be members of other sets.YES: {John, Paul} ϵ { { }, {John}, {Paul}, {John, Paul} }NO: {John} ϵ {John, Paul, George, Ringo }
Subsets
We say that A is a subset of B when all of A’s members are members of B. ( x)(x ∀ ϵ A → x ϵ B). We write: A B. Examples: ⊆
{0, 1} {0, 1, 2}⊆{0, 1} {0, 1}⊆{0, 1} {1, 0}⊆{0, 1} { x | x is a number}⊆{ } {0, 1}⊆
Subsets: More Examples
{1} {1}⊆1 1⊄1 {1}⊄{1} {{1}}⊄
Axiom of Extensionality
For any sets A and B:
A = Bif and only if
A and B have the same members( x)(x ∀ ϵ A ↔ x ϵ B)
A B & B A⊆ ⊆
Extensionality Examples
{0, 1} = {0, 1}{0, 1} = {1, 0}{0, 1} = { x | x is a natural number & x2 = x}{1} = {1, 1, 1}
The Empty Set
The empty set is the set with no members. There is only one and it is a subset of every set.
The Empty Set is a Member of Every SetProof:Let a be an arbitrary object. Since { } has no members, it follows that
a { }∉Additionally, (P → Q) is true whenever P is false. So:
a ϵ { } → a ϵ BFor any set B. Since a was arbitrarily selected:
( x)(x ∀ ϵ { } → x ϵ B){ } B⊆
There is Only One Empty Set
Proof:Suppose A and B are sets with no members.Then A B (from previous proof).⊆And B A (same proof).⊆So A = B by the Axiom of Extensionality.
Further Notation
Sometimes { } is written: ∅
Set Theoretic Operations
Let A and B be sets.
Union:A B = { x | x ∪ ϵ A or x ϵ B}
Intersection:A ∩ B = { x | x ϵ A and x ϵ B}
Comparison of Laws
(A B) C = A (B C)∪ ∪ ∪ ∪(A ∩ B) ∩ C = A ∩ (B ∩ C)
A B = B A∪ ∪A ∩ B = B ∩ A
A A = A∪A ∩ A = A
A ∩ (A B) = A∪A (A ∩ B) = A∪
(A v B) v C ↔ A v (B v C)(A & B) & C ↔ A & (B & C)
A v B ↔ B v AA & B ↔ B & A
A v A ↔ AA & A ↔ A
A & (A v B) ↔ AA v (A & B) ↔ A
Comparison of Laws
A { } = A∪A ∩ { } = { }
A (B ∩ C) = (A B) ∩ (A C)∪ ∪ ∪A ∩ (B C) = (A ∩ B) (A ∩ C)∪ ∪
A v (P & ~P) ↔ AA & (P & ~P) ↔ (P & ~P)
A v (B & C) ↔ (A v B) & (A v C)A & (B v C) ↔ (A & B) v (A & C)
Power Sets
For any set A, A’s power set is defined as follows:
POW(A) = { x | x A }⊆
If A has N members, then POW(A) has 2N members. That’s why it’s called a power set. Sometimes people write 2A to denote POW(A).
POW({0, 1, 2})
0 1 2{0, 1, 2} Y Y Y
{0, 1} Y Y N{0, 2} Y N Y
{0} Y N N{1, 2} N Y Y
{1} N Y N{2} N N Y{ } N N N
Russell’s Paradox
The Naïve Comprehension Schema
Basic idea of set theory:
When you have some things, there is another thing, the collection of those things.
For every predicate F: (Ǝy)( x)(x ∀ ϵ y ↔ Fx)
Bertrand Russell
• One of the founders of analytic philosophy (contemporary Anglophone philosophy).• One of the greatest logicians of
the 20th Century• Showed that the basic idea of
set theory can’t be right.
Russell’s Paradox
Consider the predicate:
~x ϵ x
Russell’s Paradox
According to Comprehension:
(Ǝy)( x)(x ∀ ϵ y ↔ ~x ϵ x)
Let’s call “y” here “R” for Russell’s Paradox Set.
Russell’s Paradox
R = { x | ~x ϵ x }
Question: R ϵ R?
Russell’s Paradox
Let’s suppose: R ϵ R. Then:
R ϵ { x | ~x ϵ x }~R ϵ R
Russell’s Paradox
Let’s suppose: ~R ϵ R. Then:
R ϵ { x | ~x ϵ x }R ϵ R
Russell’s Paradox
The Naïve Comprehension Schema leads to a contradiction.
Therefore it is false.
There are some properties with no corresponding set of things that have those properties.
Fixing Set Theory
Regularity
Part of our problem seems to arise from the weirdness of sets that can have themselves as members. So we can legislate that away:
Axiom of Regularity:
Restricted Comprehension
That doesn’t solve the paradox though!
Q: Why?
Restricted Comprehension
New strategy: start with the basic elements and then specify explicitly which sets exist. Instead of:
For every predicate F: (Ǝy)( x)(x ∀ ϵ y ↔ Fx)
We have:
For every predicate F: ( z)(Ǝy)( x)(x ∀ ∀ ϵ y ↔ Fx & x ϵ z)
Axioms of Set Theory
Pair Axiom: For any x and y, the set { x, y } exists.Union Axiom: For any sets A and B, A B exists.∪Power Set Axiom: For any set A, { x | x A } exists. ⊆
There are more axioms, but you get the point.
The Universe of Sets
Are we happy with this solution?
Cantor’s Diagonal Proof
Numbers vs. Numerals
Decimal Representations
A decimal representation of a real number consists of two parts:
A finite string S1 of Arabic numerals.
An infinite string S2 of Arabic numerals.
It looks like this:
S1 . S2
We can’t actually write out any decimal representations, since we can’t write infinite strings of numerals.
But we can write out abbreviations of some decimal representations.
1/4 = 0.251/7 = 0.142857
π = ?
_______
We will prove that there cannot be a list of all the decimal representations between ‘0.0’ and ‘1.0’.
A list is something with a first member, then a second member, then a third member and so on, perhaps continuing forever.
Choose an Arbitrary List
1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …
2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …
3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …
4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …
5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …
6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …
7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …
8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …
…
Find the Diagonal
1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …
2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …
3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …
4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …
5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …
6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …
7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …
8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …
…
Diagonal = 0.85263096…
Add move each numeral ‘1 up’– so ‘8’ becomes ‘9’, ‘5’ becomes ‘6’, etc.
New Representation = 0.96374107…
New Number Not on the List
‘9’ ‘6’ ‘3’ ‘7’ ‘4’ ‘1’ ‘0’ ‘7’ …
1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …
2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …
3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …
4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …
5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …
6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …
7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …
8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …
…
Doesn’t Help to Add It In!
‘9’ ‘6’ ‘3’ ‘7’ ‘4’ ‘1’ ‘0’ ‘7’ …
1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …
2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …
3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …
4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …
5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …
6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …
7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …
8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …
…
Discussion Questions
• Does this prove you can’t list all the real numbers?• How do we fix the proof?• Can you use a similar proof to show that the rational numbers aren’t
countable?• Can you list the powerset of the natural numbers?