This is not the place to give a complete introduction to transfinite arithmetic. However, let me try to give you a sense of what's happening. You would agree

This is not the place to give a complete introduction to transfinite arithmetic. However, let me try to give you a sense of what's happening. You would agree that a collection of three cars is not the same as a collection of three cats, and yet you would agree that 3 equals 3. Just because we are counting different objects, that does not mean that the number of objects is necessarily different. Specifically, we regard two collections as being the same size if we can put them in one-to-one correspondence with nothing left over. When we are talking about the size of a collection, it doesn't matter what is in the collection. Similarly when we talk about the collection of positive even numbers and the collection of positive whole numbers. The collections are different, but it's possible to put each into one-to-one correspondence with the other. In this sense we feel that it's natural, right and appropriate to say that the two collections are the same size. Using the same reasoning we can show that the collection of positive whole numbers (all of them) is the same size as the collection of positive even numbers. We pair off each number with its double, and that gives us a one-to-one correspondence. Given that such a correspondence exists it would be perverse to claim that the collections are of different sizes. The size of each is the same "infinity", even though the objects themselves are different. And so we proceed. We can show that all sorts of collections are (in some cases surprisingly) the same size. Primes, even integers, rationals, algebraic numbers, all are collections that can be put into on-to-one correspondence. Having become comfortable with that, it then comes as a shock to discover that there are collections that have infinitely many objects, but which cannot be put into one-to-one correspondence. If you try to pair up the real numbers with the positive whole numbers you always have some reals left over. Always. The generally accepted conclusion is that the collection of real numbers is genuinely bigger. And so to your closing question. Suppose we (quite reasonably) define X-1 as "take a collection of size X, remove one object, and X-1 is the number of objects left." If you start with an infinite set (any infinity, and any example) and remove one element, then the result can be put into one-to-one correspondence with the original. This can be proved. It's not hard, but it's a bit icky, and I won't do it here. But under reasonable definitions of the terms, yes, I can provide a proof that infinity-1=infinity. I hope that helps.