Thinking about infinity

[Alex K]

...  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...

...  Consider the coastline problem.  ...  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...

...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.

This is very helpful, and I do hope Alex K can help out at his convenience. I see a few different aspects in play here. Please correct any of the following:

1) The current models of the universe [may?] exclude the possibility of an actual infinity of things, but as models, these are subject to being replaced by better models and new evidence.

2) I found the coastline problem very helpful. Even though the accuracy with which you can measure the coastline's length can be improved ad infinitum, the length itself being measured is approached (and perhaps never reached) as a finite boundary. The process of increasing accuracy may be considered notionally infinite (like a geometric series?), while the coastline's length remains finite.

3) The infinite divisibility of a line is similar to the coastline problem. The "process" of identifying (counting) more and more points on that line may proceed ad infinitum. The infinity of points on a line are notionally infinite, rather than an infinity of points all simultaneously being the length of the line? <= This one is still a bit fuzzy.

Someone left the Bat signal on?

1) We have currently no evidence that the universe is spatially finite, or vice versa. I don't think it is in principle possible to scientifically rule out that it is finite.

2) The coastline problem as I know it is that it looks like a fractal at intermediate scale, and therefore its measured length depends on a certain power of the resolution of the measuring apparatus. Formally, the mathematical model for fractal coastlines yields an arbitrary large length for finer and finer resolution. In reality, the fractal is cut off when stuff is made from quantumç

3) I don't understand the question