I was not discussing the size - number of elements in the infinite set. I was discussing the value of both infinities. The first 'one to one' element is larger than the first element of that second set called infinity. 1 ≠ 2. Second item in each set: 2 ≠ 4 . This continues into infinity. In each case - each one to one correspondence - the two infinities (∞) will never be equal because one set starts with 1 and the other starts with the larger number 2.

In which case ∞ + 1 which does equal ∞ actually defines two different sets - both called ∞.

Meanwhile, what is the answer to Shocker's 'cool math trick'. Where is the overlooked restriction in his algebra?

SERIOUSLY.pull my finger!

I now see the subtle mistake in Shocker's 'cool math trick'.
Given S={1, 2, 4, 8 , 16...} and 2S={2, 4, 6, 8, 16, 32 ...},
then 2S + 1 = {1, 2, 4, 8, 16, 32 ...} IOW the infinite set called S and the infinite set called 2S has one less element than the infiinite set called 2S + 1. Yes they are all equal to infinity. But in each case, infinity has a different value. In the case of 2S + 1, the infinity also has one more element. Therefore we have equated infinities that are actually different.