The power set of the power set of an infinite set

[julypraise]

Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?

 

[chiro]

Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?

Isn't that just a set with 2^n elements (the power set)? If so couldn't you just write P(P(S)) where S your set in question? The set itself would be uncountable but that shouldn't change the properties of the result should it?

Do you have a particular set S in mind or are you just interested in any uncountable set for S?

 

[julypraise]

Because X is countabliy infinite, you can take any set that is countably infinite such as Q or Z^+ or Z, etc. And as most people know, P(X) in this case is equivalent to R. But my question here is what on Earth P(R) for example? Is it an, like, example of large cardinal?

 

[chiro]

Because X is countabliy infinite, you can take any set that is countably infinite such as Q or Z^+ or Z, etc. And as most people know, P(X) in this case is equivalent to R. But my question here is what on Earth P(R) for example? Is it an, like, example of large cardinal?

I think this might be of use to you:

http://en.wikipedia.org/wiki/Beth_number#Beth_two

 

[julypraise]

Thanks, it was indeed helpful!

 

[micromass]

There are some natural examples of sets which such a cardinality.
An important example (although quite advanced), is the Cech-Stone compactification of $\mathbb{N}$. This can be shown to have the same cardinality of the powerset of $\mathbb{R}$.

This is NOT an example of a large cardinal. Large cardinals are very large, and can not be written so easily as the powerset of R. Indeed, the existence of large cardinals can usually not be shown, only postulated.

 

[ImaLooser]

Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?

If you mean physical objects, then there are no infinite sets on Earth.

 

[ImaLooser]

Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?

The power set of an infinite set is always of a higher cardinality than that of the infinite set.