What is the first cardinal number which is bigger than cardinality of the set of all natural numbers?

Answer 1

The previous poster got it wrong.

By definition, aleph_1 is the next cardinal after aleph_0, which is the cardinality of the naturals. The continuum hypothesis says that this is the same as the cardinality of the set of real numbers (or the set of all subsets of the naturals). It is known that the continuum hypothesis is independent of the Zormelo-Fraenkl axioms for set theory. In fact, there is a result due to Paul Cohen that states that there are models of ZFC in which the cardinality of the reals can be any aleph_k where k is any ordinal which is not of countable cofinality.

Sorry, the previous poster clearly misunderstood their own source. The alephs are defined in terms of successor cardinality. So aleph_1 is the next cardinality after aleph_0. The beths are defined in terms of the power set operation, so beth_1 is the cardinality of the reals. The question at issue in CH is whether beth_1=aleph_1. BY DEFINITION, aleph_1 is the next cardinality after aleph_0. But beth_1 may not be. The generalized continuum hypothesis (GCH) says that beth_alpha=aleph_alpha for all ordinals alpha.

As for ZFC, it certainly cannot deal with CH (or GCH), but it also cannot deal with large cardinal axioms. *Any* axiomatization will fail to deal with *some* question. Woodin and company are trying to find suitable *additional* axioms that will settle CH. Whether their axioms are all that intuitive is a matter of opinion, but I tend to think not.

Added:
I actually prefer GCH since it resolves cardinal arithmetic in one axiom. But that is just my taste. The ultimate resolution will come from which axioms give the most aesthetic mathematics.

Answer 2

This is a trick question.
A first KNOWN cardinal is called "aleph 1" which is equivalent to a power-set of all integers(a set of all subsets of integers).
However, it impossible to prove or disprove the existence of smaller cardinal then aleph 1, yet bigger then aleph null(the size of all naturals).[1]
The question is much debated because it seemingly indicates deficiencies in ZFC set of axioms for set theory.

P.S.
Concerning 2nd answer:

No, I got it right.

ZFC IS independent of continuum hypothesis, and THEREFORE it is under fire as a bad axiomatization.
Theoretically a GOOD axiomatization would DEAL with continuum hypothesis.

NOW aleph 1 was THOUGHT by Cantor to be the cardinality immediately greater then aleph null BECAUSE Cantor thought that continuum hypothesis is true.
In fact you can phrase the continuum hypothesis as this:

"aleph 1 is the first cardinal greater then aleph 0"

But since it is presently neither true nor false there COULD be a cardinality in between aleph null and aleph 1 if we come up with some axiomatization that would prove continuum hypothesis false.

P.P.S
Ok, Ok!
I take it all back! the next poster is correct.
I need to brush up on my set theory.

I however do not take back my criticism of ZFC.
Sure according to Godel a complex enough axiomatization will create some statements that are neither provable nor disprovable, but CH SHOULD not be one of those statements.
Is not it time to DROP ZFC and work from scrtach. I mean we can use a new axiom to fix ZFC: "CH is true." But this is a cop out.