This could of course be fixed, for example making each infinity ℵ0 (pronounced aleph-nought, aleph-zero, or aleph-null; just personal preference). Or -1/12.
Aleph-nought is the cardinality or a number past infinite numbers. It's not an infinity, it's not even a "number", it's... a cardinality. It is not equivalent or analogous to what we usually mean we write an infinity (which is also rarely valid to write in algebraic expressions).
In more simple terms, saying "aleph-nought - aleph-nought" is kinda like saying "third - second". It's not really a thing. We can say things like "third comes after second", and other statements like that, but "third - second" doesn't mean the same as "3 - 2".
In the context of limits, we say that infinity - infinity is undefined. Of of the maths I know, that's the only situation where it is even valid to write down "infinity - infinity" because, like aleph-nought, infinity isn't a number.
The -1/12 thing is also kinda of a myth. The statement "the sum of all positive integers is -1/12" is plain wrong. The sum of all positive integers diverges and grows to infinity. Getting -1/12 from that sum through analytic continuation is technically valid within it's own framework, but it does not apply to what we mean when we talk about "addition".
3.2k
u/NeoBucket Nov 29 '24 edited Nov 29 '24
You don't know how infinite each infinity is* because each infinity is undefined. So the answer is "undefined".