Yeah, the cardinals are really mind blowing -- but only as long as you consider them to be "real", that is, if you hold a Platonic view on reality and consider mathematical objects (that is, sets) to really exist. This naive dream is quickly crushed by introductory model theory, where you learn that there are exists a set 'S' (a lot of them actually) which can be regarded to be "the set of all sets",and a special relation 'e' between the elements of this set which represents a set membership. Together, the set and the relation form a structure we call a model of set theory. All true statements about sets are also true, if we restrict ourselves to our model and replace the usual set membership relation with our special relation.

So, in our model 'S' there is some element 'o', such that there is no 'x' in 'S' such that 'x e o' -- that is, no 'x' is in a relation 'e' with 'o'. If we take into account that relation 'e' is supposed to represent set membership in our model, we notice that this object 'o' represents an empty set in our model, the existence of one is ensured by usual set theory axioms. Since every object in the set theory is a set, so is our 'o', and although it represents an empty set in our model, it does not need to be in fact empty -- it is enough that it is empty in the sense of our special membership relation 'e'.

Apart from "empty set" 'o', in our model there are representatives for all the usual sets we know -- the set of natural numbers, reals, functions from naturals to {1, 2} (and {1, 2} as well) -- and under our relation 'e' they behave in exactly the same way as usual sets behave under usual set membership relation.

So far, our models looks exactly like the whole universe, only smaller. But, thanks to some results of model theory (precisely, the Lowenheim-Skolem theorem), we can impose another restruction on our model -- we can require it to be countable.

Now this is really mind blowing -- our model behaves just like the whole universe and yet there is only as many elements in it as there are natural numbers! Sounds quite paradoxical -- you could ask, but what about Goedel's theorem? It sure has to hold, because this is a model of set theory, but yet there are exactly as many reals as naturals in our model -- countably many. How is that even possible?

Well, the answer is quite obvious -- the concept of "cardinality" is not absolute. When we say that two sets have the same cardinality, we mean that there is a bijection between their elements. Since our model is countable, there is a bijection between elements representing natural numbers and the ones representing reals. But there is no paradox -- this bijection is not an element of our model. It shows that that naturals' and reals' representatives are in bijection, but only in the whole universe. When we restrict ourselves to our model an we ask for an element representing such a bijection, none exists. While there are exactly as many 'reals' as there are 'natural numbers' in our model, we cannot see it from inside.

Now, maybe this is the case with our regular numbers -- they are the same in number, but it is impossible for us to see. But the longer one think about this problem, the less sense the question makes. The simplest solution is to accept the fact that sets have no real existence whatsoever, that the mathematician does not explore structure of some abstract constructs, but only manipulates the strings on paper in some defined way. This approach is not very romantic, but is the only way I know of escaping from problems and paradoxes brought by Platonic view on reality.