[criloz2]

This feels really arbitrary to me, I am more found to think that exist two additional disjoint categories just unbounded and bounded, and we can have unbounded countable sets and bounded countable sets, and an order relation between the cardinalities of unbounded countable sets just don't exist, it doesn't make any logical sense. In which point you stop walking both sets to compare their size?

[mcguire]

You don't actually walk the sets to compare their sizes. You establish a 1-1 relationship between their elements.

Take the natural numbers 0,1,2,3... (call that N) and the even counting numbers 0,2,4,6... (call that E).

Here's a function between N -> E: f(n) = n * 2. You can prove that f maps every element of N to an element of E, and that every element of E is mappable from an element of N. f is 1-1. And therefore, N and E are the same "size".

[brianpan]

An integral can be used to find the area under a curve by summing infinitesimally small areas.

The point isn't to walk to the end of the infinity. It's to use it as a tool to calculate something in a different or previously impossible way.

All tools are arbitrary without an application. But sometimes you have to figure out what's possible before you can try to find an application.

[Kranar]

That's one interpretation but it's kind of shaky without a theory of infinitesimals, and when you do work with a theory of infinitesimals it turns out not to be all that much better than working with the more traditional delta/epsilon approach, which is really quite rigorous and can be explained without the use of any infinities.