Thread: Bounded Comprehension Principle

Using the Bounded Comprehension Principle, show that if x and y are sets then
so are:

a.

b.

c.

I'm just not sure the Bounded Comprehension Principle applies here... could someone explain this?

Let me try the first one:



(to answer the question you only need the first equation)

I guess I wasn't clear in my description

You have to demonstrate that for all of the sets

Originally Posted by DrSteve

Let me try the first one:



(to answer the question you only need the first equation)

What is P(x)? My definition of Bounded Comprehension is for every property phi(x) and every ordinal a, the set {x: rk(x) <a and phi(x)} exists at time a

P(x) is the Power Set of x, that is .

Since x is a set, so is P(x) (by the power set axiom).

I realize that you need to do all of them. I did the first one to start you off. If you understand this one, then you should at least be able to attempt the other two. Give them a try, show your thoughts, and I'll help out if you get stuck.

I don't understand how this is using the bounded comprehension principle

The Comprehension schema

Bounded Comprehension is where is a set.

Originally Posted by DrSteve

The Comprehension schema

Bounded Comprehension is where is a set.

Bounded Comprehension is for every property and every ordinal , the set exists at time

this is my definition and I'm not sure how it relates to yours

I've never heard that definition, but it is equivalent to the one I have given you. For example, if has rank , the rank of can be expressed in terms of (I think it has rank ). Conversely you can find a bounding set for a collection of sets of rank less than a fixed ordinal.