It has long been established that there are a hierarchy of infinities, in which one infinity is greater than another. Does this carry over into our understanding of physics, and if so how? What is the lowest hierarchy of infinity? The counting numbers? How many hierarchies of infinities are there?
Typically divergent answers are signs that you're making a mistake. For example, in QFT, you typically get divergent answers when you're integrating over loops. But this is just a sign that you are integrating over unrealistic regions of phase space, and you have to pick some way to not do that. So I don't think we make a distinction between infinities in physics. A counterexample is, of course, comes from string theory. Placing d-branes at various types of topological singularities can give very different effective field theories.
Interesting question because I just came to the forum to ask basically the same thing. The thread http://www.sciforums.com/showthread.php?t=89936 regarding two entities choosing random rooms in an infinite hotel analyzed this concept and raised the following question for me: are we so sure that infinite-length irrational numbers "exist" but the "countably infinite" set of naturals contain no such infinite-length numbers? The interpretation of this difference is crucial when discussing infinities and I believe Cantor's bijection argument hinges upon a very specific one. Before any mathematicians have an aneurysm consider this: represent the set of naturals in "base 1" where the progression is 1, 11, 111, 1111, 11111...now, using the Axiom of Infinity to define the Naturals I feel it's possible to claim that it is a contradiction to say that this set is truly infinite but none of its members are. By definition there are infinitely many members in this set, but the each member has a digit size equal to its ordinal value in that set...can the infinite size of the set ever be "satisfied" without simultaneously satisfying the infinite size of its members? Someone help!
OK, you have to keep the container separated from the contents. When you claim that a countably infinite set of numbers exists, you make a space for them (a container); you do this with your set of base1 quantities, for example. Then, you assume a generator that 'counts out' numbers in base1. This process is the infinite part, the numbers are all finite and the "machine" can always generate another number.
So the "infinite" in the infinitely countable is just that the generator always has the ability to make another member? It has nothing to do with the number of members existing in the set? I'm still not comfortable with this logic. I feel after you define a set then its members exist whether they are enumerated or not.
Well, you induce the set, starting with 1. You count out 1, 11, 111, ... so countability is a recurrence. Then you induce that the counting can continue indefinitely. After an indefinite number of countings you have the numbers, not when you start with 1 in base1. Is the countability infinite by induction? The counting process if it's infinite means the set, at any given count is "countably infinite", by induction.
Don't we? Aren't some infinities in QFT "sick" infinities and others not? For example, a logarithmic divergence can be dealt with but a quadratic divergence cannot.
I think you're getting mixed up with renormalisability. If the theory is renormalisable then any divergence can be renormalised away. For non renormalisable theories this is not possible. Also, I know of at least one theory that contains quadratic divergences but is still renormalisable and I'm sure there are many more.
The various infinities of Cantor's ordinal numbers are distinct from those of Cantor's cardinal numbers, which are distinct yet again from those found in real and complex analysis. I believe that the infinities in quantum field theory are more analogous to those of analysis, and usually result when a theory is pressed beyond it's realm of applicability. What's promising about quantum field theory is that renormalization indicates that while the question as asked may produce a nonsense result, there is a question which represents a better way of asking the same question that gets a finite result. We've got about 4 people with experience teaching undergraduates who can tell me if I have murdered or done otherwise irreparable violence to the idea.
That's what I meant: There are two qualitatively different types of infinities in QFT. There are those you can renormalize and those you can't. OK, bad example on my part.
The infinities themselves aren't really different in either case. For renormalisable theories it means you've made a calculational error as explained very well by Ben in post #2 and for non renormalisable theories it just means that the theory is never physical. The infinities are not different per se, but in one case you can get rid of them. A better way of stating this is that one theory allows you to ask a more physical question that can't be asked in the other theory.
