You have run into one of the most obvious respects in which infinite collections of things behave differently than finite collections. (Incidentally, why should they behave similarly in
any respects?)
There are some respects in which they behave in exactly the same way: e.g., the definition of "is the same size as" (i.e., has a 1-to-1 correspondence with") works equally well for finite and infinite sets. But the most obvious definition of "is larger [smaller] than" which works for finite sets won't work for infinite sets. One finite set S is larger than another finite set S' iff the members of S' can be paired 1-to-1 with the members of a proper subset of S. So {a,b,c,d,e} counts quite properly and intuitively as larger than {1,2,3}. {a,d,e} is a proper subset of {a,b,c,d,e}; and {1,2,3} can obviously be paired with the former. That's all it takes to show that one finite set is larger (or, conversely, smaller) than another.
But if "larger than" and "smaller than" were thus simply defined for infinite sets, paradoxes would arise. The standard set-theoretic definition of these notions for infinite sets is as close to the familiar definitions for finite sets as we can get without lapsing into paradox. In fact, the following definition works just fine for both finite and infinite sets:
Any set S (whether finite or infinite) is larger than another set S' iff *they are not the same size*, but S'
is the same size as a proper subset of S. (By the bye, if you don't like that definition of "is larger than" for infinite sets -- the one that causes problems for you -- then I challenge you to propose a different one that works.
Interestingly, Galileo recognized that there is the same number of positive integers and even positive integers, even though in some sense there are "twice as many" of the former as the latter.
An infinite set has denumerably many members if it makes sense to think of them all as coming as a first and a second and a third and a fourth and a fifth and so on (without repetition)
ad infinitum. Given that (i.e., a pairing with the positive integers), all of your sets are denumerable and hence of the same size. (At the finite as well as the infinite level, the relation "is the same size as" is transitive: if collection A is the same size as B, and B the same size as C, then A is the same size as C.)
For both more detail and more fun, see <
http://www.unconventional-wisdom.com/WAW/INFINITE.html>;
There'll be a quiz at the end of the period.