Countable versus uncountable infinities in math and physics

aikiddo

In math and physics, one often takes the limit of an expression involving an integer N as N → ∞, and ends up with the expression of a continuous variable x. Some examples of this are:

- An integral as the limit of a Riemann sum of N terms
- A string with continuous mass density as the limit of a string with N discrete masses
- A path integral as the limit of an integral over N independent variables

The limits in each of these cases give countable infinities -- an infinite number of discrete beads on a string is countable -- but we seem to treat them as uncountable, such as when taking an integral over the real line.

Clearly treating the limits this way gives the correct answer, but what allows us to treat a countable infinity as an uncountable one in these cases, or is there a gap in my reasoning somewhere?

mathman

I believe the answer is the same for all your examples. I'll describe it for Riemann sum. Each term in the sum is the area of a rectangle. The base of the rectangle is a line segment with an uncountable number of points. So each term in the sum involves an uncountable number of points - therefore it is not surprising that going to the limit gives an integral over a real interval, with an uncountable number of points.

aikiddo

Alright, I agree, but I don't see how that applies to the other examples. Point masses on a string are discrete; you can think of the mass density before taking the limit as the sum of delta functions, and then taking the limit replaces the sum with an integral.

The same sort of thing happens when one goes from a set of countably infinitely many degrees of freedom to a continuous field in statistical mechanics. In neither case is the sum Riemannian, yet we are justified in approximating it with an integral.

voko

What happens here is that even though the point masses are discrete, the sections of the string they are attached to are continuous. That's what becomes "dx" in the integral.

Generally, every integral must contain "dx" in some form. And the finite sum it is the limit of must have a corresponding "Δx". Without that, you don't have an integral. And this is what connects countable N with the continuum.