Can infinity be defined rigorously?

[dipole]

Can "infinity" be defined rigorously?

In math, many of the most fundamental theorems and structures rely on the concept of an infinity.

For example, we define the irrational (and transcendental) numbers as being the limits of some Cauchy sequence of rational numbers. This limiting process inherently assumes that you can continue the sequence "to infinity", but it seems to me we never really quite define exactly what we mean by doing that. The idea of an integral is another example, which is essentially an infinite series.

The number zero, which is sort of like infinity has algebraic properties. When we want to know what we mean by "zero", we can define it as being the element of some set such that,

$\{ x \in S: x + \phi = x \forall x \} , \{ x, \phi \in S: x*\phi = \phi \forall x \}$

However, as far as I know, infinity has no algebraic properties. It's simply something that sequences tend towards when they fail to converge. Yet, we seem to treat it concretely when actually looking at the limits of such sequences.

It seems like it's almost used to mean different things, depending on the context. What is the rigorous foundation for the idea of infinity? Why should you even be able to take a limit in the first place? Is this simply just a postulate of mathematics?

 

[Jorriss]

However, as far as I know, infinity has no algebraic properties. It's simply something that sequences tend towards when they fail to converge. Yet, we seem to treat it concretely when actually looking at the limits of such sequences.

Look up the extended real number line.

 

[arildno]

As Joriss implies, there lots of number systems in which "infinity" is regarded as a number, with its own algebraic properties.

However, you seem to confuse this with how, for example, the concept of a limit is formulated.

When we say that Some A_n goes to A as n goes to infinity, that is short hand for saying that for any chosen e>0, we may find some number M so that the inequality n>M directly IMPLIES |A_n-A|<e

 

[eigenperson]

Infinity does mean different things in different contexts; there are a number of different definitions.

When you deal with limits of sequences, you do deal with infinite sets. An infinite set is easy to define; it's a set which can be put into one-to-one correspondence with a subset of itself. The existence of infinite sets is indeed an axiom (the so-called "axiom of infinity", which is actually a little stronger than that). Most people find it a very intuitive axiom, since they believe there should be such a thing as "the set of natural numbers," and such a set can only be infinite. People who don't agree that we should accept the axiom of infinity are called finitists, and there are a few of them out there -- but only a few.

However, the "number" ∞ never appears in the definition of limit, nor do you ever have to work with it in order to understand limits. Take a look at the definition of limit and you'll see that there is no "infinity" anywhere in it. Of course, mathematicians frequently use the symbol ∞ as shorthand, but that's informal.

 

[Mandelbroth]

This limiting process inherently assumes that you can continue the sequence "to infinity", but it seems to me we never really quite define exactly what we mean by doing that. The idea of an integral is another example, which is essentially an infinite series.

First, limits assume nothing about infinity. When we talk about limits, we are talking about being able to make something arbitrarily close to something, to put it vaguely.

Second, don't get used to seeing integrals as special cases of infinite series. It turns out that the integral is the more general operation, and you'll learn later on in mathematics that summations can actually be seen as special cases of integrals, using the works of a guy named Lebesgue. Your definition of the integral tends to change drastically as you learn more mathematics. Eventually, you might see that integrals act on differential forms, which allows everything you know about integrals to be summed up in 2 formulas.

 

[johnqwertyful]

"Can infinity be defined rigorously" yes, that's the entire field of analysis.