1. Naïve Sets & Russell’s Paradox
Philosophical Tools
1st Term 2016
Dr. Michael Johnson

2. Instructor Dr. Michael Johnson You can call me Michael
If you must, Dr. Johnson

3. About Me From San Antonio, TX, U.S.A.
Undergraduate degree (B.A. philosophy): University of Texas- Austin (2003).
Graduate degree (Ph.D. philosophy): Rutgers, The State University of New Jersey (2011).
Moved to Hong Kong August

4. Tower of Saviors: IGN: Mj UID: 67599983 OpenRice UN: mjhk
Find Me
Tower of Saviors: IGN: Mj UID: OpenRice UN: mjhk

5. Course Website
michaeljohnsonphilosophy.com > PHIL 2000 Philosophical Tools 2016

6. Instructor

7. Course Textbook The course textbook is available:
In the bookstore (???)
In the library
As a free Ebook in the library

8. The Universe of Sets

9. How Many Things?

10. Mereology Theory of parts and wholes
Are there bigger things than particles?
Arbitrary fusions
Nihilism?

11. How Many Things?

12. Lots of Little Things…

13. Some Weird Things

14. One Maximal Thing

15. 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

16. How Many Things?

17. How Many Things?

18. How Many Things?

19. How Many Things?

20. How Many Things?

21. How Many Things?

22. How Many Things?

23. History of Set Theory Founded by Georg Cantor in 1847.
Popular ever since.

24. 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?

25. Set Theoretic Operations
Let A and B be sets. Union: A ∪ B = { x | x is in A or x is in B} Intersection: A ∩ B = { x | x is in A and x is in B}

26. 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 A A & B ↔ B & A A v A ↔ A A & A ↔ A A & (A v B) ↔ A A v (A & B) ↔ A

27. Comparison of Laws
A ∪ { } = A A ∩ { } = { } A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A v (P & ~P) ↔ A A & (P & ~P) ↔ (P & ~P) A v (B & C) ↔ (A v B) & (A v C) A & (B v C) ↔ (A & B) v (A & C)

28. Russell’s Paradox

29. The Naïve Comprehension Schema
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)

30. 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.

31. Russell’s Paradox
Consider the condition: x is not a member of x

32. 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”)

33. 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)

34. Russell’s Paradox
R = { x | x is not a member of x } Question: Is R a member of R?

35. 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

36. 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

37. 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.

38. 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?