Sean, one problem is that people can’t follow the arguments you suggest without these things being made explicit. So I’ll try to do that:
Suppose the badness of distributed dust specks approaches a limit, say 10 disutility units.
On the other hand, let the badness of (a single case of ) 50 years of torture equal 10,000 disutility units. Then no number of dust specks will ever add up to the torture.
But what about 49 years of torture distributed among many? Presumably people will not be willing to say that this approaches a limit less than 10,000; otherwise we would torture a trillion people for 49 years rather than one person for 50.
So for the sake of definiteness, let 49 years of torture, repeatedly given to many, converge to a limit of 1,000,000 disutility units.
48 years of torture, let’s say, might converge to 980,000 disutility units, or whatever.
Then since we can continuously decrease the pain until we reach the dust specks, there must be some pain that converges approximately to 10,000. Let’s say that this is a stubbed toe.
Three possibilities: it converges exactly to 10,000, to less than 10,000, or more than 10,000. If it converges to less, than if we choose another pain ever so slightly greater than a toe-stubbing, this greater pain will converge to more than 10,000. Likewise, if it converges to more than 10,000, we can choose an ever so slightly less pain that converges to less than 10,000. If it converges to exactly 10,000, again we can choose a slightly greater pain, that will converge to more than 10,000.
Suppose the two pains are a stubbed toe that is noticed for 3.27 seconds, and one that is noticed for 3.28 seconds. Stubbed toes that are noticed for 3.27 seconds converge to 10,000 or less, let’s say 9,999.9999. Stubbed toes that are notice for 3.28 seconds converge to 10,000.0001.
Now the problem should be obvious. There is some number of 3.28 second toe stubbings that is worse than torture, while there is no number of 3.27 second toe stubbings that is worse. So there is some number of 3.28 second toe stubbings such that no number of 3.27 second toe stubbings can ever match the 3.28 second toe stubbings.
On the other hand, three 3.27 second toe stubbings are surely worse than one 3.28 second toe stubbings. So as you increase the number of 3.28 second toe stubbings, there must be a magical point where the last 3.28 second toe stubbing crosses a line in the sand: up to that point, multiplied 3.27 second toe stubbings could be worse, but with that last 3.28 second stubbing, we can multiply the 3.27 second stubbings by a googleplex, or by whatever we like, and they will never be worse than that last, infinitely bad, 3.28 second toe stubbing.
So do the asymptote people really accept this? Your position requires it with mathematical necessity.
Sean, one problem is that people can’t follow the arguments you suggest without these things being made explicit. So I’ll try to do that:
Suppose the badness of distributed dust specks approaches a limit, say 10 disutility units.
On the other hand, let the badness of (a single case of ) 50 years of torture equal 10,000 disutility units. Then no number of dust specks will ever add up to the torture.
But what about 49 years of torture distributed among many? Presumably people will not be willing to say that this approaches a limit less than 10,000; otherwise we would torture a trillion people for 49 years rather than one person for 50.
So for the sake of definiteness, let 49 years of torture, repeatedly given to many, converge to a limit of 1,000,000 disutility units.
48 years of torture, let’s say, might converge to 980,000 disutility units, or whatever.
Then since we can continuously decrease the pain until we reach the dust specks, there must be some pain that converges approximately to 10,000. Let’s say that this is a stubbed toe.
Three possibilities: it converges exactly to 10,000, to less than 10,000, or more than 10,000. If it converges to less, than if we choose another pain ever so slightly greater than a toe-stubbing, this greater pain will converge to more than 10,000. Likewise, if it converges to more than 10,000, we can choose an ever so slightly less pain that converges to less than 10,000. If it converges to exactly 10,000, again we can choose a slightly greater pain, that will converge to more than 10,000.
Suppose the two pains are a stubbed toe that is noticed for 3.27 seconds, and one that is noticed for 3.28 seconds. Stubbed toes that are noticed for 3.27 seconds converge to 10,000 or less, let’s say 9,999.9999. Stubbed toes that are notice for 3.28 seconds converge to 10,000.0001.
Now the problem should be obvious. There is some number of 3.28 second toe stubbings that is worse than torture, while there is no number of 3.27 second toe stubbings that is worse. So there is some number of 3.28 second toe stubbings such that no number of 3.27 second toe stubbings can ever match the 3.28 second toe stubbings.
On the other hand, three 3.27 second toe stubbings are surely worse than one 3.28 second toe stubbings. So as you increase the number of 3.28 second toe stubbings, there must be a magical point where the last 3.28 second toe stubbing crosses a line in the sand: up to that point, multiplied 3.27 second toe stubbings could be worse, but with that last 3.28 second stubbing, we can multiply the 3.27 second stubbings by a googleplex, or by whatever we like, and they will never be worse than that last, infinitely bad, 3.28 second toe stubbing.
So do the asymptote people really accept this? Your position requires it with mathematical necessity.