| > What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem? Although our mathematical systems rely on the existence of computable procedures (that halt) to e.g indicate whether a formula is well formed or verify a proof, they have no trouble talking about uncomputable functions. There actually ends up being a hierarchy, as once you allow "solve the halting problem for this Turing machine" as an operation the expanded set of computable functions is now unable to solve its own halting problem: https://en.wikipedia.org/wiki/Arithmetical_hierarchy > These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders. These kind of things are actually pretty well studied. Some interesting examples are: * The hyperreal numbers which include infinitely many distinct numbers which are infinite or infinitesimal: https://en.wikipedia.org/wiki/Hyperreal_number * The surreal numbers which include all ordered fields as a subfield (reals, complex numbers, hyperreals, etc.): https://en.wikipedia.org/wiki/Surreal_number * The quaternions and octonions, which along with complex numbers are the only finite-dimensional algebras over the real numbers that can have "division" and "absolute value" in the usual sense: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi... In general, mathematicians are really good at investigating things of the form "Okay we have structure X with properties A, B, C. What if we get rid of C? How about B? How about B and a weaker version of A? ...". |