Set theory

[Fredrik]

I'm thinking about buying Goldrei (http://www.amazon.com/exec/obidos/tg/detail/-/0412606100/) or Potter (http://www.amazon.com/exec/obidos/tg/detail/-/0199270414). Both books are getting excellent reviews at Amazon, especially Goldrei.

I would like to learn about the ZFC axioms, cardinals and ordinals, etc. I assume both books will cover those topics. I'm also very curious about something I heard for the first time today:
...the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation.


Am I reading this right? Every set in ZFC theory can be constructed from the empty set by iterating the power set operation?! That sounds weird. I know you can define the integers this way, and infinite ordinals I guess, but everything else?

0={}
1={0}
2={0,1}
...

If this one axiom leads to all the ZFC axioms, then I should have heard about this before, so I feel that I must have misunderstood something.

Do both of these books explain these things?

[Hurkyl]

Let V denote the class of all sets that are contained in some transfinitely-iterated power of the empty set. (As in the Wiki article)

Suppose the set S0 is not in V. Then S0 must contain an element, S1, not in V. Iterating this argument (and using the axiom of choice), we can construct an infinite chain \cdots \in S_2 \in S_1 \in S_0. This contradicts the axiom of foundation, therefore S0 cannot exist.



Now, this fact does not imply all the axioms of ZFC.

This fact cannot tell you "how many" ordinal numbers are available for use -- the class of ordinals is, roughly speaking, one of the "inputs" to this construction.

This fact cannot tell you much about what kinds of sets exist and how complex they can be. e.g. it cannot tell you whether or not there exist sets whose cardinality lies between those of Vw+1 and Vw.

This fact cannot even tell you that two sets are equal iff they have the same members!

[Fredrik]

Now, this fact does not imply all the axioms of ZFC.

If "this fact" refers to the idea that all sets are constructed by iterating the power set operation, then you have convinced me that it can't replace the ZFC axioms (probably not even a single one of them). You saved the best argument for last:


This fact cannot even tell you that two sets are equal iff they have the same members!
I should have realized that myself. I guess I didn't really think because I started the thread to ask what book I should get, and ended up throwing in a question that seems silly now.

Let V denote the class of all sets that are contained in some transfinitely-iterated power of the empty set. (As in the Wiki article)

Suppose the set S0 is not in V. Then S0 must contain an element, S1, not in V. Iterating this argument (and using the axiom of choice), we can construct an infinite chain \cdots \in S_2 \in S_1 \in S_0. This contradicts the axiom of foundation, therefore S0 cannot exist.

I don't understand this. I'm not sure I understand the power set construction either, but it seems to me that the first steps are

V0={}
V1={{}}={0}
V2={{},{0}}={0,1}
V3={{},{0},{1},{0,1}}={0,1,{1},2}
V4={0,1,{1},{2},2,{0,{1}},{0,2},{1,{1}},{1,2},{{1} ,2},3}

Suppose your V is V4. You said that if S0 isn't in V, it must contain an element that isn't in V. But {0,{1}} isn't in V (it's a subset, but not a member) and it doesn't contain anything that isn't in V. So your proof seems to fail at the first step.

But I have no doubt that you'll explain what I have misunderstood and make me feel silly again. :smile:

[Hurkyl]

If "this fact" refers to the idea that all sets are constructed by iterating the power set operation,
The (von Neumann) universe is constructed by iterating the power set operation. The theorem is that all sets are elements of the universe.



As for your example, the set you chose is a member of V4, so I don't understand what you were getting at.

Incidentally, the contrapositive of my claim is (hopefully) more obvious:
If every element of S is in V, then S is in V

This requires all of V -- if you stop iterating at some point, then it breaks down. For example, a set whose elements all lie in V4 is not necessarily a member of V4. (But it is necessarily a member of V5)

[Fredrik]

Sorry about the {0,{1}}. How about {0,{2}}? That's not in V and doesn't contain anything that isn't in V.

Also, my notation is inconsistent. I should have written 0,1,2,... on the left-hand sides, or V0,V1,V2,... on the right-hand sides. It doesn't make much sense to mix two notations the way I did.

[Hurkyl]

Sorry about the {0,{1}}. How about {0,{2}}? That's not in V and doesn't contain anything that isn't in V.
Equivocation fallacy. When I made that statement, I was using V to refer to the entirety of the Von Neumann universe. But you're using V to refer to the V4 subclass.

[Landau]

The book by Potter is besides a mathematical book also a philosophical book. I guess this wouldn't be the best choice for learning the subject for the first time.
You may also want to take a look at Suppes (http://www.amazon.com/Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304/ref=sr_1_1?ie=UTF8&s=books&qid=1253393159&sr=1-1).

[Fredrik]

Equivocation fallacy. When I made that statement, I was using V to refer to the entirety of the Von Neumann universe.
OK, that wasn't at all clear, I think. But now that you've cleared that up, your proof is easy enough to understand. I needed some help from Wikipedia to understand why that sequence contradicts the axiom of foundation, but I think I got it now.

[Fredrik]

The book by Potter is besides a mathematical book also a philosophical book.

That doesn't seem like a bad thing. But the reviews at Amazon suggest that Potter is using some "new" approach to the subject. It doesn't seem to be about ZFC at all, but rather about some other set theory. If that's right, it's certainly not the best place to learn about ZFC, but I might still be interested in the things he is talking about. Have you read this book? Can you tell me what it's about?

[Fredrik]

The (von Neumann) universe is constructed by iterating the power set operation. The theorem is that all sets are elements of the universe.

I'm still :confused: about this. If we define V0={}, V1={V0}, V2={V0,{V0}}={V0,V1}, and so on, Vn is a set with 2n-1 members for n=1,2,3,... If these are the only sets we allow, there are no sets with 3 members. Is {1,2,3} not a set?!

[Hurkyl]

I'm still :confused: about this. If we define V0={}, V1={V0}, V2={V0,{V0}}={V0,V1}, and so on, Vn is a set with 2n-1 members for n=1,2,3,... If these are the only sets we allow, there are no sets with 3 members. Is {1,2,3} not a set?!
{1,2,3} is an element of the universe because it's an element of V5.

V is not the class whose elements are the Va: V is the class which is the union of all the Va.