Posted: Dec 26, 2017 4:23 am
I remember an explanation I once heard of the concept of the convergence of a sequence to a limit. The idea is that a sequence converges if for any specified error margin epsilon then there is some index number for which all subsequent terms of the sequence after that index number are within the error margin of the value the sequence converges to. This is likened to a call and response challenge, if someone doubts that a sequence converges, then they are challenged to provide some value of epsilon where this can't be done - and then the response is to specify the index number that meets the challenge.
There's a very similar, but much more boring version of that game with negative integers. For any point in the set, it's really not that hard to specify how many times +1 has to be added to get back to zero. Or for an integer set indexing days in the past how many days must pass from any day within an infinite past to get to the present.
It is a peculiar feature of certain strains of thought amongst (for want of a better term) mathematical laymen that somehow fails to distinguish between integers and an affine extension of the integers and that declares the winning move in the game is to ask how far -∞ is from 0, and to consistently overlook the fact that -∞ isn't an integer.
There's a very similar, but much more boring version of that game with negative integers. For any point in the set, it's really not that hard to specify how many times +1 has to be added to get back to zero. Or for an integer set indexing days in the past how many days must pass from any day within an infinite past to get to the present.
It is a peculiar feature of certain strains of thought amongst (for want of a better term) mathematical laymen that somehow fails to distinguish between integers and an affine extension of the integers and that declares the winning move in the game is to ask how far -∞ is from 0, and to consistently overlook the fact that -∞ isn't an integer.