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]

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.

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.

 

[CellCoree]

As a scientist, it is important to understand the concept of infinity in both math and physics. In mathematics, infinity is often treated as a concept rather than a number, allowing us to express and manipulate infinite quantities in a meaningful way. In physics, infinity can represent the limit of a physical quantity, such as the speed of light or the size of the universe.

In the examples given, the concept of taking a limit as N approaches infinity allows us to bridge the gap between countable and uncountable infinities. While the number of discrete beads on a string may be countable, the limit of an infinite number of beads allows us to consider the string as having a continuous mass density, which can be treated as an uncountable infinity.

In essence, the concept of limits allows us to extend the idea of counting to infinity, allowing us to consider infinite quantities as continuous and uncountable. This is a fundamental concept in both math and physics, and it is crucial in understanding and solving complex problems.

However, it is important to note that this treatment of infinity is not without its limitations. In some cases, such as the study of fractals, we encounter infinities that cannot be treated as continuous or uncountable. In these cases, we must use different mathematical tools to understand and describe these infinities.

In conclusion, the concept of taking a limit allows us to treat countable infinities as uncountable in certain cases, providing us with a powerful tool in understanding and solving problems in both math and physics. It is a fundamental concept that continues to be studied and refined by scientists and mathematicians alike.