how do we assume there is infinity?

Question

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?

Answer 1

Mathematics, being a human creation, doesn't necessarily have anything to do with the universe (besides which, I would hardly say we know the universe is not infinite).

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." - J. W. N. Sullivan

"Mathematics is a game played according to certain simple rules with meaningless marks on paper." - David Hilbert

We can take anything as an axiom that we want, though of course we tend to focus on the collections of axioms that are interesting to us. In particular, there is nothing preventing us from taking as axioms statements that do not describe what (we think) we know about reality, and different people may well decide to study the consequences of different collections of axioms - even if those collections contradict each other! You are entirely welcome to study set theory where it is taken as an axiom that infinite sets do not exist, as I'm sure people already do.

Answer 2

There is $1$ and there is $n+1$. That's infinity.

Answer 3

Note that while it's tempting to think that mathematics is only used to model our physical reality, this is not true.

If this was true, then what sense does the number $10^{100}$ make? It's larger than the number of particles in the visible universe, so surely we can't represent it physically.

And yet, even the ancient Greek believed that if $n$ is an integer, then $n+1$ exists. So if $10^{100}$ doesn't exist, but for every $n$ which exists, $n+1$ does exist... something goes wrong.

Even if you don't talk about infinite sets, infinity is inherent into the natural numbers as we are used to thinking about them. Sets were created to allow collections of mathematical objects (like numbers) to be mathematical objects on their own accord. So naturally, we are inclined to talk about the set of natural numbers which is infinite.

Some people do reject this approach to mathematics, they may believe that infinite sets do not exist, but there are infinitely many natural numbers nonetheless; or sometimes that there is a largest number (even though we don't know what it is). These philosophical (and mathematical) schools of thought are joined under the term "finitism" (and ultrafinitism in the latter case).

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Some threads of interest.

1. What does it mean for a set to exist?
2. Is math built on assumptions?
3. What good is infinity?

Answer 4

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).