What is the cardinality of the set of infinite cardinalities?

Question

I am currently aware of only two infinite cardinalities:

• $\aleph_0 = |\Bbb N|$
• $\aleph_1 = |\Bbb R|$

Questions:

1. Is there an infinite number of infinite cardinalities?
2. If yes, is this set of cardinalities countable or uncountable?

I know that $\aleph_0<\aleph_1$ and I would tend to guess that the set of infinite cardinalities is countable at most (or possibly even finite), since there is no known element $K$ such that $\aleph_0<K<\aleph_1$.

BTW, are there any other acceptable operations (such as $+,-,\times,/$) between elements in this set?

Thanks

Answer 1

Firstly, kudo's for being interested in the cardinal numbers. However, there's a few errors in the question, so lets just try to set the record straight.

The (currently standard) collection of set-theoretic principles is called ZFC. So rather than asking "what is true of the cardinal numbers?" (vague question), let us ask "what can ZFC prove about the cardinal numbers?" (precise question).

ZFC proves the following sentences.

1. $\beth_0 = |\mathbb{N}|$
2. $\beth_1 = |\mathbb{R}|$ (but we cannot prove $\aleph_1 = |\mathbb{R}|$).
3. $\beth_0 < \beth_1$.
4. There exist infinitely-many infinite cardinal numbers.
5. There does not exist a set of all cardinal numbers.

See also, beth numbers, aleph numbers.

To see why there ought to exist infinitely-many infinite cardinal numbers:

Step 1. Recall Cantor's theorem: $|X| < |\mathcal{P}(X)|$ for all sets $X$.

Step 2. Consider the following sequence. $$|\mathbb{N}|, |\mathcal{P}(\mathbb{N})|, |\mathcal{P}(\mathcal{P}(\mathbb{N}))|,...$$

(Of course, this is just the initial portion of the beth sequence: $\beth_0, \beth_1,\beth_2,\ldots$)

I might have a go answering some of your other questions a bit later.

Answer 2

The "set" of cardinalities is not a set. It is a class. Every set is a class, but some classes are sets and some aren't. There are things that you can make with a set and not with a class, and one of these things is defining its cardinality. But a class that is not a set always has infinitely many elements.

So:

1. There is no "set of cardinalities". Thus, there is no cardinality of the set of cardinalities. I'm not sure, but this sounds like the typical thing that drives to paradox; perhaps the cardinality of the sets of cardinalities would be bigger than any other cardinality (that would be the reason that it doesn't exist).
2. There are infinitely many cardinalities. A simple proof has been written by JiK in his/her comment.

Look here for some details.

Answer 3

Q1.yes.there is not only an infinite number of infinite cardinalities,but the collection of infinite cardinals is a proper class,that is too big to be a set,bigger than all cardinals. Q2.it is surely uncountable,and also bigger than all uncountable cardinals. aleph-1=cardinality of R is true if and only if CH is true,otherwise R can have cardinality aleph-2,aleph10,aleph-1232337312,or other alephs.cardinals have opertion + and *,a+b=a*b=max{a,b} for all infinite cardinals.- and / can not be defined. you can have unimaginable big alephs,such as aleph-aleph-aleph-.......,aleph-0 times,that is the smallest cardinal such that aleph-a=a.BUT it is consisitent with ZFC,it is still smaller than the cardinality of R,2^aleph-null!!!aleph-x just means the xth infinite cardinal.