The Axiom of the Power Set

honestrosewater

I'm going to tack a question on here in case anyone comes along to comment. I thought it was more that the "uncountable" set contained in the countable model was, from the outside, a countable set, and it was only uncountable within the model because the model contained no bijection between that set and N. Hm, that doesn't really make sense as I say it. I have cleared this up with myself before, but the clarifications I have tried have apparently not stuck.

matt grime

That sounds familiar - it is uncountable in the sense of 'there does not exist a bijection with the inductive set (i.e. N, or whatever it should be called) within the model' but there is a bijection in some extended model.

Of course - Cantor's 'diagonal' method works for any set to show there is no bijection between X and P(X) - assume f is any injection from X to P(X), and define S:={x in X : x not in f(x)}, then S is in P(X) and S is not in Im(f).

So I was talking nonsense, earlier.

dmuthuk

Now, how exactly can we use the power set axiom to argue that for any "collection" or class of sets, there exists a set that contains them all as elements? Is this even possible or does this lead to a paradox? We already know that we can construct this set for a finite collection without the power set axiom.

Hurkyl

If you could, do you see how to construct a set of all sets? (And thus a contradiction)


A sample use of the power set axiom is, given two (possibly infinite) sets A and B, to construct the set of all functions from A to B.

Another use is to construct the set of all unary relations on A.

Can you see how the power set axiom is equivalent to the statement

The class of all functions from A to B is a set?

The power set axiom is the reason why set theory is interesting. Without it, we might as well just be doing logic.


Have you seen first-order logic yet? What about second-order logic?

dmuthuk

So, I assume [tex] f:A \to \{0,1\} [/tex] is an indicator function for a subset of [tex]A[/tex] that we have in mind (i.e. a way to sort out which elements are in our subset and which aren't).

I wish Halmos' book talked more about "classes". I actually picked this book up because I started questioning the validity of existence theorems in my analysis and algebra courses, and I thought set theory would answer some of these questions, and it does straighten some of these ideas out.

honestrosewater

What do you want to know about them? Did you read about limitation of size?

Here is my understanding. I could be mistaken, of course, so take it with a grain of salt.

Suppose that you start with some vague concepts of objects and membership. You want objects to be able to contain other objects as members, and you would ideally like to treat all objects as the same type with respect to membership. But it turns out that this causes problems, and you need to have at least two types of objects. So you rename your objects, or whatever you've been calling them, "classes", and you call your two types of classes "sets" and "proper classes".

Now you have some options about how to remake your theory of sets. You know one thing for sure: proper classes can't be members of sets. So you can push proper classes completely aside and pretend like they don't even exist as far as your set theory is concerned. This, I think, is basically what ZF does. You can also allow both sets and proper classes into your set theory and work in the rule that proper classes cannot be members of sets (by modifying the definition of the membership relation, introducing a limitation of size axiom, etc.). This is what some other set theories do, e.g., NBG.

P.S. Yes, I'm pretty sure the indicator/characteristic function is what Hurkyl had in mind.

fopc

Too many posts to read, but I can comment on your opening post.

Powerset axiom: You picked on the wrong axiom to derive from the others. Try deriving pairing axiom from the rest.

Flaw in argument? It went bad in sentence two.
Specification doesn't give you license to say that.

Belonging:
A primitive binary relation on the set under discussion U and its powerset P(U).
That makes it a subset of U x P(U).

Equality:
He merely says its a binary relation between sets "perhaps"
more elementary than belonging. Well belonging
is taken as a primitive, so what could be more
elementary than a primitive? (I think he was just joking.)
Really what I think he's saying is that equality is essentially identity, and that everything is made all right by the underlying predicate logic that comes equipped with identity. If you track it down, I think in the end you'll find Leibniz's law. Anyway he (Halmos) uses '=' freely from then on.

For Halmos, no distinction is made between a collection of sets and a set of sets, or a family of sets (ch.9) for that matter.

dmuthuk

If I understand the Specification axiom correctly, it says that given a set A, and some logical sentence that doesn't necessarily have to depend on the members of A, then those members of A for which this sentence is true is a set. Now, the way I understand the word "subset" is simply a "bunch" of elements that are also members of A and that satisfy some logical sentence, which by Specification, is a set. So, I am not sure where I went wrong with the second sentence.

fopc

"Let E be a set. Each subset x of E exists by the axiom of specification."

Specification (loosely by ZF) says for such an (arbitrary) E, there exists a set B [given any X (X member of B) iff
(X member of E and P(X))].

It doesn't say anything about the existence of each (i.e., every) subset x of E.
It's an axiom schema. How many axioms? One for each predicate. Suppose E=N (set of natural numbers). Now how many subsets? But you feel specification declares the existence of all subsets of any arbitrary set (N in this case). This would result in how many axioms?