Is the cardinality of a set necessarily a natural number? I've never seen phrases like "$\sqrt{5}$ people" or "a set with $\pi$ many elements". Are there sets with cardinality, say, $\frac{1}{2}$?
Edit: As Brian M. Scott pointed out, the only real numbers that are cardinalities of sets are the non-negative integers. Could you explain why is it so?
 A: Fundamentally, cardinals and real numbers are different things.  You can think of the cardinality of a set as some abstract object ("cardinal") assigned to it, in such a way that two sets get assigned the same cardinal if and only if there is a bijection between them.
But there is a natural way to identify the finite cardinals with the natural numbers: namely, identify a cardinal $a$ with the natural number $n_a$ such that the set $\{1, 2, \dots, n_a\}$ has cardinality $a$ (i.e. any other set with cardinality $a$ has a bijection with $\{1,2,\dots, n_a\}$ and any other set with cardinality $a$).  (In this discussion, "natural numbers" includes 0.)  This is a nice identification because it makes cardinal arithmetic match up with the arithmetic of natural numbers.  For instance:


*

*Ordering: $n_a \le n_b$ if and only if any set of cardinality $a$ has an injection into any set of cardinality $b$

*Addition: If $A,B$ are disjoint and have cardinalities $a,b$ respectively, then the cardinality $c$ of their union $C = A \cup B$ has $n_c = n_a + n_b$.

*Multiplication: If $A,B$ have cardinalities $a,b$ and $C = A \times B$ has cardinality $c$, then $n_c = n_a n_b$.

*Exponents: If $A,B$ have cardinalities $a,b$, and $C = A^B$ is the set of all functions from $B$ to $A$, then $n_c = n_a^{n_b}$.
One could imagine a system that identified other (infinite) cardinals with real numbers other than the natural numbers.  For instance, nobody could stop me from proposing a system in which the cardinality $\aleph_0$ of the set of integers is identified with the real number $22/7$.  But this system wouldn't have the properties listed above.  For instance, $22/7 \le 4$, but there is no injection from $\mathbb{Z}$ to $\{1,2,3,4\}$, and $\mathbb{Z} \times \{1,2,3,4,5,6,7\}$ definitely does not have the same cardinality as $\{1,2,\dots, 21,22\}$.  In fact, it's not hard to see that there would be no way of identifying infinite cardinals with real numbers that would preserve the above list of properties.
The properties above are very useful, and so in order to preserve them, we generally do not attempt to identify any other real numbers with cardinals.  In principle, there could be another system that, although it didn't satisfy the above properties, had some other useful properties.  But I've never heard of one that was useful enough to attract much attention.
A: I am copying Brian M. Scott's comment here, because I think it answers the question completely and correctly:

As the term cardinality is normally used, the only real numbers that are cardinalities of sets are the non-negative integers. The cardinalities of infinite sets, of course, are not natural numbers.

