I don't want to get between the fight here. But as to the point of whether we need infinity in the "real" world:
I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.
On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line. It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation. Also if we don't have continuum, then we have polynomials without roots etc. And also transcendental numbers like pi and e which seem to have real world implications in physics etc wouldn't exist. Although they are finite numbers, you can never really reach them, only approximate them. But if they don't exist in the real world, then how can you have a disk of radius exactly 1, which would then have an area of exactly pi, if you don't have exactly pi.