Infinities are okay if they come with a definition of convergence. For example, we can say that an infinite sequence of real numbers x1, x2, x3… “converges” to a real number y if every interval of the real line centered around y, no matter how small, contains all but finitely many elements of the sequence. For example, the sequence 1, 1⁄2, 1⁄3, 1⁄4… converges to 0, because every interval centered around 0 contains all but finitely many of 1, 1⁄2, 1⁄3, 1⁄4… Some sequences don’t converge to anything, like 0, 1, 0, 1..., but it’s an easy exercise to prove that no sequence can converge to two different values at once.
Now the only sensible way to understand 0.999… is to define it as whatever value 0.9, 0.99, 0.999… converges to. But that’s obviously 1 and that’s the end of the story for people who understand math.
You can use the same procedure for infinite sums. x1+x2+x3+… can be defined as whatever value x1, x1+x2, x1+x2+x3… converges to. For example, 1+1/2+1/4+1/8+… = 2, because the sequence of partial sums is 2-1, 2-1/2, 2-1/4, 2-1/8, … and converges to 2.
By now it should be clear that 1+2+3+4+… doesn’t converge to anything under our definition. But our definition isn’t the only one possible. You can make another self-consistent definition of convergence, where 1+2+3+4+… will indeed converge to −1/12. But that definition is complex, esoteric and much less useful than the regular one, which is why that viral video really shouldn’t have used it without remark.
Most paradoxes involving infinity are just pulling a fast one on you by not specifying what they mean by convergence. If you try to use the common sense definition above, or really any self-consistent way to assign values to infinite expressions, the paradoxes usually go away.
Infinities are okay if they come with a definition of convergence. For example, we can say that an infinite sequence of real numbers x1, x2, x3… “converges” to a real number y if every interval of the real line centered around y, no matter how small, contains all but finitely many elements of the sequence. For example, the sequence 1, 1⁄2, 1⁄3, 1⁄4… converges to 0, because every interval centered around 0 contains all but finitely many of 1, 1⁄2, 1⁄3, 1⁄4… Some sequences don’t converge to anything, like 0, 1, 0, 1..., but it’s an easy exercise to prove that no sequence can converge to two different values at once.
Now the only sensible way to understand 0.999… is to define it as whatever value 0.9, 0.99, 0.999… converges to. But that’s obviously 1 and that’s the end of the story for people who understand math.
You can use the same procedure for infinite sums. x1+x2+x3+… can be defined as whatever value x1, x1+x2, x1+x2+x3… converges to. For example, 1+1/2+1/4+1/8+… = 2, because the sequence of partial sums is 2-1, 2-1/2, 2-1/4, 2-1/8, … and converges to 2.
By now it should be clear that 1+2+3+4+… doesn’t converge to anything under our definition. But our definition isn’t the only one possible. You can make another self-consistent definition of convergence, where 1+2+3+4+… will indeed converge to −1/12. But that definition is complex, esoteric and much less useful than the regular one, which is why that viral video really shouldn’t have used it without remark.
Most paradoxes involving infinity are just pulling a fast one on you by not specifying what they mean by convergence. If you try to use the common sense definition above, or really any self-consistent way to assign values to infinite expressions, the paradoxes usually go away.