>>12323079The mathematician shrugs, for he is confident that either it does converge to zero or it does not, even though he cannot demonstrate either.
So in conclusion, to COPE with not being able to actually sum every infinite series they want to mathematicians have:
- Invented a completely new number system whose elements are formal INFINITE SEQUENCES which are DEFINED by these series and identifications which are in general completely indeterministic and in most cases open problems. This means that whatever calculations you were able to do in the naturals or rational numbers, applying algorithms to arrive at the answer, are no longer valid because for the vast majority of questions about the reals there is NO ALGORITHM, no process to get to an answer.
- Declared the answer of each infinite series which satisfy the precious condition which they think should ensure that the series converges to be ITSELF as a formal infinite sequence (leaving aside the questions of whether or not such a sequence can actually be completed).
- In the process completely fucked up all the size/cardinality issues in mathematics because nobody knows how big this fucking object they created is to the point that there are now mathematicians who think that asking how big it is is not even a sensible question (even though it's completely sensible when you're talking about actual deterministic objects like natural numbers or rational numbers).
- Fucked up the field of algebra which used to be finitistic and deterministic by using this monstrosity of an object in the proofs to the point where it's hard to tell which parts of algebra are finitistic and deterministic (not reliant on this new number system) and which are not.
