One would like to present a consistent view of mathematics. Often, however, the mathematics that one is involved with at the moment tends to colour the view. For example, I can currently teaching two courses: (a) Discrete mathematics and (b) The theory of computation. The enclosed view of the role of infinity in mathematics is clearly shaded by this coincidence.
Numbers play multiple roles:
- As a way of counting. (Cardinals)
- As a way of ranking. (Ordinals)
For natural (finite) numbers, these different senses of the use of numbers coincide. (Though one can have some doubts when the numbers are really large!)
In the same way, mathematical objects can be considered in different ways:
- "Pure" set-theory. (Axiomatic set theory)
- Sets with some sort of structure. (Category theory)
Even for finite sets, these notions do not coincide! For example, there are many different finite groups of the same size.
Coming to the problem of "infinity". The simplest notions of infinity are:
- The set of natural numbers. (Cardinal aleph_0)
- The ordered set of natural numbers. (Ordinal small omega)
- Asymptotic points or points at infinity. (For example, the point (1:0:0) in projective geometry)
Each of the above have associated arithmetic and algebraic operations. For example, with counting numbers we have addition and as a consequence multiplication. With ordinal numbers we have the notion of a successor which can be used to define a notion of addition. The corresponding structure in sets is that of Boolean or sigma algebra of sets. Category theory also has its own notion of algebra called ``universal algebra'', which is like (but not quite the same as) the sigma algebra of sets (infinite sums and products need to be defined and may not exist!).
So to re-phrase the question, we are asking if the ordinary notion of arithmetic and algebraic operations extends to infinity.
At first glance it does. We can certainly perform Boolean operations with infinite sets. The problem is that the usual statements about these operations are sometimes no longer true and our intuition about algebraic identities would fail us.
For example, it is usual to say that multiplication is the operation of repeated addition. When the number of additions is infinite, it is not very evident what this means. We define the product of sets AxB which clearly explains what this operation (multiplication) is for sets.
Similarly, it is natural to think of addition as repeated successor operations, but it is not always clear what this means for the infinite successor operation. Again, ordinal succession is defined in a way that such an operation is meaningful through the notion of limit ordinals.
However, in each case some "obvious" results from the finite case are no longer valid.
It is worthwhile to extend notions from the finite to the infinite when this is useful in giving us expectations regarding questions (about finite sets!) that we could not have arrived at otherwise. (For an interesting example, have a look at the Goodstein sequence.)
As an addendum, I would like to add the naive re-statement of Skolem-Lowenheim.
Since language consists of countably many sentences, we can only hope to define countably many things and from a practical point of view we can only define finitely many things.
Thus, infinity is a notion that mathematicians handle with care, limiting the roles that it can take, so that playing around with infinity gives meaningful (and correct!) results about the finitely many things that we will actually encounter! Mastering this way of handling infinity with care is what a lot of mathematical training is about.