I heard an expressions,
some infinities are larger than others recently, and they stated that it was proved to be so. I haven't been able to find this proof, and I'm not as much of a math enthusiast as I was earlier in my life, but I don't believe this statement to be true. I also do not know how to format the math symbols here, so forgive me if I'm a bit vague.
Given a set of numbers S1 = [0, 1], we can say that there are an infinite amount of numbers in that set. If the previous statement,
some infinities are larger than others is true then we can also say
some infinities are smaller than others and also
some infinities are equal in size to others. With this in mind, we could say that if S2 = [1, 2], then there are just as many numbers as there are in S1, as there are in S2, because the range is the same. If both of the previous statements are true, than we could define S3 = S1 + S2, which would yield all numbers in the set [0, 2], and we could say that S3 is larger than both S1 and S2, since it contains all numbers in S1 and S2. With this in mind, we could also say that the set [0, 0.1] has 1/10 as many numbers (but still infinite) as S1.
This is where I lose understanding. What this is saying is that we can have infinity == infinity + infinity == infinity / 10. But because the law of identity states that $A = A$, then how can
some infinities are larger than others be true? My understanding is that infinity will always equal infinity, and there are just as many numbers between $[0, 0.1]$ and $[0, 1000]$. If infinity does not equal infinity, then it violates the law of identity. By saying
some infinities are larger than others it seems to me like we're saying
infinity is the set of all numbers, and there exists a set whose size is larger than infinity.
Am I wrong in this understanding?