Thinking about infinity

[TheRealJoeFish]

First, it may be true that there can only exist a "finite number of things" (whatever that means) in a universe, but maybe for a different reason than the obvious one: there may be a lower limit on the "size" of something, and thus a limit on the number of things that can be in a given place. Expressed a different way - and, as always when I'm talking about physics, I ask Alex K to elaborate if what I'm saying makes sense or correct me if it doesn't - if there's a finite amount of energy in the universe, and a lower limit on the smallest "amount" of energy that can be transferred at a given time, that may necessarily constrain the number of "things" that can exist. I'm thinking Planck lengths and scales and such, based on my rudimentary understanding of quantum physics mumbo jumbo.

From my (far more rigorously cultivated) mathematical perspective, I think it's important when discussing "infinity in the real world" to distinguish between what I'll call "actual infinity" and "arbitrary largeness." For example, consider the coastline problem. If you try to measure the coastline of an island with a yardstick, you'll get a certain answer for the length of the coastline. Because the yardstick doesn't bend, though, you'll be missing lots of tiny ins and outs that are less than a yard in size. So, when you measure the coastline with a ruler, you'll get a bigger number. If you measure the coastline with rope, you'll get a bigger number still, because you'll be lining up the bendy rope with the edge of the water, but you'll still miss all of the little wiggles in the coastline that are smaller than the width of the rope. You'll get a bigger size if you measure with string, and bigger still if you measure with fishing wire (but you'll still be missing the myriad of microscopic wiggles smaller than the width of the fishing line, such as the contours of grains of sand). What this means is, you'll never measure the coastline to be "infinity", but, if you pick any length, there's a level of precision at which the measurement of the coastline will be greater than that length.

Or, maybe a better example, with the line: Sure, you may never be able to count an "infinite" number of points on a line. But, if I give you any number at all (like, 52 billion), with enough time you could count more points on the line than that number.