Mathematicians who are finitists.

[Smaug]

Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?

It's a subtle distinction.   Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also.  However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets.  And, so, some infinities are bigger than others.  The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.

What I meant is that if we consider all sets to be fundamentally finite as finitists suggest then there's no need for such a term as cardinality. You can just explicitly specify the number of elements for any given set. And I still can't grasp the finitists' point of view, how they beat the "+1" arguement. For me such recursive definition automatically leads to the notion of infinity. Infinity makes perfect sense from practical point of view, too. In a non-rigorous approach it can be interpreted as a value which is out of scale.