Questions about infinity

[Steven G]

I have been thinking about and researching the concept of infinity.
Here is what I understand.
The Natural Numbers are said to be a countable infinite set. I understand how to map other countable infinite set to the naturals.
I understand how to prove that the interval (0,1) is uncountable making the Reals an uncountable infinite set.
I concluded from that since R = Q u Q', that Q' must be uncountable.
Given a countable infinite set A, I understand the proof that shows that P(A) is uncountable. I need to think of why P(P(A)) is a different infinity but suspect the proof will be similar to going from A to P(A).
If I can abuse the less than symbol, I now know that A<P(A)<P(P(A)) < .... That is, there are an infinity number of different infinities

Now I have a couple of questions. Does the reals have the same cardinality as one of the infinite infinities I mention above. If yes, can you tell me which one?
Are there other infinities? After seeing what happens with power sets my gut feeling is yes.
What are some open questions regarding infinities?

 

[topsquark]

I have been thinking about and researching the concept of infinity.
Here is what I understand.
The Natural Numbers are said to be a countable infinite set. I understand how to map other countable infinite set to the naturals.
I understand how to prove that the interval (0,1) is uncountable making the Reals an uncountable infinite set.
I concluded from that since R = Q u Q', that Q' must be uncountable.
Given a countable infinite set A, I understand the proof that shows that P(A) is uncountable. I need to think of why P(P(A)) is a different infinity but suspect the proof will be similar to going from A to P(A).
If I can abuse the less than symbol, I now know that A<P(A)<P(P(A)) < .... That is, there are an infinity number of different infinities

Now I have a couple of questions. Does the reals have the same cardinality as one of the infinite infinities I mention above. If yes, can you tell me which one?
Are there other infinities? After seeing what happens with power sets my gut feeling is yes.
What are some open questions regarding infinities?

Call the cardinality of a set A, |A|.

The cardinality of the reals is equal to the cardinality of the powerset of the counting numbers, ie. [math]| \mathbb{R} | = | P( \mathbb{N} )| = | 2^{| \mathbb{N} | } |[/math]. The "continuum hypothesis" states that there is no infinite cardinal between [math]| \mathbb{N} |[/math] and [math]| \mathbb{R} |[/math]. It has not been proven.

Otherwise, yes, there is a list of infinite cardinals constructed from the counting numbers: [math] \aleph _0 = | \mathbb{N} |[/math], [math]\aleph _1 = |P( \mathbb{N} ) |[/math], [math]\aleph _2 = | P(P ( \mathbb{N} ) ) |[/math], etc. However, continuum hypothesis aside, I don't know if this list is inclusive.

-Dan

 

[pka]

Here is very clear discussion of the continuum problem.

 

[Steven G]

Here is very clear discussion of the continuum problem.

Very nice discussion. Thanks so much.