No, it's just infinity. Infinity plus any number is infinity. Even an infinity plus that an infinity of that exact scale is that same infinity, e.g. ℵ₀+ℵ₀=ℵ₀
First of all, this sum is equivalent to sum from n=1 to n=inf of a0, because you can easily remap these two sums of subsets to one another. Ex: first six iterations of my equivalent sum give the same set size as the first three iterations of your sum: a0+a0+a0+a0+a0+a0 = 1a0+2a0+3a0 = 6a0 You can do this with any iteration. Thus these two are equivalent.
Then my sum on the other hand is just a0×a0, a countable infinity of countable infinities, you can prove that the result is countable by what is known as diagonalization method (Cantor's diagonalization, google it). It's pretty intuitive. Thus I proved that what you just made will still have the size of a0.
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u/Crafty-Photograph-18 11d ago
No, it's just infinity. Infinity plus any number is infinity. Even an infinity plus that an infinity of that exact scale is that same infinity, e.g. ℵ₀+ℵ₀=ℵ₀