Download 03-Infinite

Infinite Sets
PHIL 2000
Tools for Philosophers
1st Term 2016
Topics for Discussion
• What are sets?
• Where are they?
• How do they relate to their members?
• Do they exist?
• How are they different from Venn diagrams?
• How many sets are there?
• Which axiom tells us this?
• How many empty sets are there?
Barber Paradox
The Barber Paradox
Once upon a time there was a
village, and in this village lived a
barber named B.
The Barber Paradox
B shaved all the villagers who did
not shave themselves,
And B shaved none of the villagers
who did shave themselves.
The Barber Paradox
Question, did B shave B, or not?
Suppose B Shaved B
1. B shaved B
Assumption
2. B did not shave any villager X where X shaved X
Assumption
3. B did not shave B
1,2 Logic
Suppose B Did Not Shave B
1. B did not shave B
Assumption
2. B shaved every villager X where X did not shave X
Assumption
3. B shaved B
1,2 Logic
Contradictions with Assumptions
We can derive a contradiction from the assumption that B shaved B.
We can derive a contradiction from the assumption that B did not
shave B.
The Law of Excluded Middle
Everything is either true or not true.
Either P or not-P, for any P.
Either B shaved B or B did not shave B, there is no third option.
It’s the Law
• Either it’s Tuesday or it’s not Tuesday.
• Either it’s Wednesday or it’s not Wednesday.
• Either killing babies is good or killing babies is not good.
• Either this sandwich is good or it is not good.
Disjunction Elimination
A or B
A implies C
B implies C
Therefore, C
Example
Either Michael is dead or he has no legs
If Michael is dead, he can’t run the race.
If Michael has no legs, he can’t run the race.
Therefore, Michael can’t run the race.
Contradiction, No Assumptions
B shaves B or B does not shave B
[Law of Excluded Middle]
If B shaves B, contradiction.
If B does not shave B, contradiction.
Therefore, contradiction
Contradictions
Whenever we are confronted with a contradiction, we need to give up
something that led us into the contradiction.
Give up Logic?
For example, we used Logic in the
proof that B shaved B if and only if
B did not shave B.
So we might consider giving up
logic.
A or B
A implies C
B implies C
Therefore, C
No Barber
In this instance, however, it makes more sense to give up our initial
acquiescence to the story:
We assumed that there was a village with a barber who shaved all and
only the villagers who did not shave themselves.
The Barber Paradox
The paradox shows us that there is
no such barber, and that there
cannot be.
Russell’s Paradox
The Axiom of Comprehension
Basic idea of set theory:
When you have some things, there is another thing, the collection of
those things.
For any condition C, there exists a set A such that:
(For any x)(x is in A if and only if x satisfies C)
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 condition:
x is not a member of x
Russell’s Paradox
According to the Axiom of
Comprehension, there exists a set
R such that:
(For any x)(x is in R if and only if x
satisfies “x is not a member of x”)
Russell’s Paradox
According to the Axiom of
Comprehension, there exists a set
R such that:
(For any x)(x is in R if and only if x
is not a member of x)
Russell’s Paradox
R = { x | x is not a member of x }
Question: Is R a member of R?
Russell’s Paradox
Let’s suppose:
R is not a member of R. Then:
R is a member of { x | x is not a
member of x }
Hence, R is a member of R
Russell’s Paradox
Let’s suppose:
R is a member of R. Then:
R is a member of { x | x is not a
member of x }
Hence, R is not a member of 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.
Questions about Russell’s Paradox
• Is there something wrong with the condition?
• Does it make sense for a set to be a member of itself?
• Does it make sense for a set to not be a member of itself?
• Must set theory be false if its basic idea is?