The observable universe is a sphere, centered on us, where the radius is the horizon distance, which is the distance light could have traveled since the Big Bang. The observable universe is finite, but is only a subset of the entire universe. Is the entire universe finite or infinite? If the universe has positive curvature, like a sphere, it could be finite. If it has zero curvature, like a plane, or negative curvature, like a hyperboloid, it is infinite. According to all our measurements, the universe is flat, and thus infinite. There is still the possibility that the universe could have positive curvature, and be finite, but where the radius of curvature is so large, that the deviation of flatness could be so small, from our point of view, that it would appear flat to us. It used to be thought that the universe might have enough mass to recollapse into a Big Crunch. This has been disproven. In fact, the expansion of the universe is accelerating. That means, we know that it will exist for an infinite length of time in the future. The Big Bang has been confirmed by the CMB, so we think of the Big Bang theory as having won the Big Bang versus Steady State debate. However, we still don't know whether the Big Bang was the fundamental beginning of time, which is the traditional view, was instead only a local Big Bang, which created this specific part of the universe, which we think of as the universe. According to eternal inflation or chaotic inflation, at any time, a given patch of space might suddenly undergo inflation. According to this view, time would extend infinitely backwards. So does infinity exist in the real universe? According to our recent theories, the universe is very probably infinite in space, definitely infinite in future time, and possibly infinite in past time. There are other occurrences of infinity in physics, such as having to sum over an infinite number of Feynman diagrams. Even elementary introductory physics requires at least simple calculus which necessarily involves the concept of infinity. When you ask, "Can you count to such and such number?", what you are asking is, does number X appear in the set of integers, Z = 1, 2, 3, …? Well, the number 1/2 also does not appear in the set of integers. Does that mean 1/2 does not exist? Why single out the integers as your set of comparison? Why not choose some other set, such as the prime numbers? Why not say the number 9 does not exist because it does not appear among the prime numbers? You can't count to infinity. You also can't count all the numbers that appear between 0 and 1. Does that imply that these numbers don't exist? You can't write down all of the digits of pi. Does that imply pi doesn't exist? In other words, infinity exists in the real universe, even if you can't count to it.