TITLE: how do we assume there is infinity?
QUESTION [9 upvotes]: Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets.
We know that our universe doesn't contain infinite number of elements (including subatomic particles), so how do we assume there is infinity?

REPLY [2 votes]: If you really mean to ask "how" we assume that there is an infinite set, that's simple: we include it as an explicit assumption (that is to say, an axiom) in the foundations of set theory.  So far, that seems to work — no-one's been able to find a contradiction between that assumption and the other usual axioms of set theory.
The reason why we need to do that is that the other standard axioms of set theory are not strong enough to let us prove it — it's possible to construct models that satisfy all the other axioms of ZF set theory, but which don't have any infinite sets.
So why do we want to assume the existence of infinite sets, then?  Well, the reason set theory was developed in the first place was so that we would have a formal language for talking about collections of numbers.  In particular, we would like to have a formal way of saying "all numbers" or, say, "all even numbers" or "all odd numbers" or "all numbers greater than 5".  In set theory, we call all those things "sets" of numbers.
Unfortunately, it's easy to show that, if a "set of all even numbers" exists, then it cannot be finite: it's possible to define a one-to-one correspondence between the set of even numbers and some proper subset of it, like, say, the even numbers greater than 10.  (Try to do it yourself!  It's not hard.)  Thus, if we want to be able to talk about "the even numbers" (or "the odd numbers", etc.) as a set, then we have to construct our set theory in such a way that it includes some infinite sets.
The alternative, of course, is to work in a theory without infinite sets, but that gets kind of awkward pretty fast, because you won't be able to formalize statements like "this property holds for all even numbers", since "all even numbers" is not a well defined set in a finitistic set theory.  You can work around such limitations in various ways, e.g. by saying "if $x$ satisfies the definition of an even number, then this property holds for $x$" instead (which formally avoids using "even numbers" as a set), but most mathematicians would prefer to avoid such logical contortions and just work in a theory in which "even numbers" is a thing (specifically, a set).