Archrono

The best argument against 0.999.... = 1 is 0.999.... will never equal 1. You must go to infinity to make a geometric series -or any other proof- prove that 1 = 0.999.... But by infinity's definition it can never be reached, therefore 0.999... never equals 1. You are never at infinity, you never have infinity in your hands to dividide by and you are never allowed to repeat a geometric series infinitely. You are treating infinity as if it is there for any fool to use. Reenforcement - As x gets really really big (goes off toward infinity, *eye roll*) 1-(1/x) represents 0.999... but x is never infinity, so it is entirely inappropriate to do arithmetic at infinity. And this is where we move on to calculus and thus math as a tool rather than a philosopher's play thing.

Every proof used has a problem. You are treating 0.999... as a discrete number and comparing it to 1. 0.999... has no value in terms of discrete numbers such as 1. How much is 0.999...? Infinitely close to 1. It is not a discrete number and is not therefore comparable. Intuition tells us that 0.999... is less than 1. Arithmetic tells us that 0.999... is equal to 1. However, logic tells us that 0.999... is not arithmetically related to 1 because it is infitecimal. The proofs apply discrete arithmetic to a non-discrete number. Now you're goin to prove that 1.000.... = 0.999... with the same silly arithmetic. Stop dividing infinitely, stop adding infinitely. You can't make apple pie from florida oranges.