chalidze
To add to my previous posting on infinity: the hope to cover a finite interval of a line by an infinite quantity of points is faulty for the reason that infinity is a quality, not a quantity.
Let's say one manages to do this (assuming, of course, that points are not put on top of each other, and there are no gaps left). The moment he declares that it is done -- that the finite interval is covered by points without gaps -- I will suggest he put one more point on that interval. But that is impossible, as there are no gaps left. If one cannot add one more
point to an infinity of points, then it is not an infinity.
Compare it with Euclides' proof that a sequence of natural numbers is
infinite: he suggested adding one more number to any quantity of
numbers. That is infinity indeed.
My conclusion: either it is impossible to cover a finite interval
with points, or else that interval contains a finite number of
points.The reader is invited to enjoy exploring this philosophical bifurcation.
Valery Chalidze. Benson, Vermont, Apr. 25, 2015
Let's say one manages to do this (assuming, of course, that points are not put on top of each other, and there are no gaps left). The moment he declares that it is done -- that the finite interval is covered by points without gaps -- I will suggest he put one more point on that interval. But that is impossible, as there are no gaps left. If one cannot add one more
point to an infinity of points, then it is not an infinity.
Compare it with Euclides' proof that a sequence of natural numbers is
infinite: he suggested adding one more number to any quantity of
numbers. That is infinity indeed.
My conclusion: either it is impossible to cover a finite interval
with points, or else that interval contains a finite number of
points.The reader is invited to enjoy exploring this philosophical bifurcation.
Valery Chalidze. Benson, Vermont, Apr. 25, 2015