John, I have a question another possible 'reality' for infinities.
Why not count motion? Bear with me...
If we don't have infinities, then we don't have irrational numbers and Xeno's paradox keeps us from ever arriving at our destination.
I realize there are multiple ... lets say routes to dealing with the resolution to Xeno's paradox but in one sense why not say that its the irrational numbers that let us arrive at a destination? - whether that destination be arriving at 1 from 0 or arriving at the finish line of a race.
I say this because since there is an infinite number of rational numbers between any two rational numbers we could have the paradox play out anytime we want to 'jump' or move anywhere. we can't step from one rational to another because of the 'gap' in between them. But once we have irrationals there is a continuity to physical space that lets us 'move', or at least prove that we did. If we were simply drawing a line with a pencil on a paper we couldn't get from A to B with only the rationals, but once we have the irrationals we can draw the line continuously. ... ok, a bit weird maybe, but could you under any stretch of the imagination by this is a 'physical' thing? (I'm also implying that this sort of knowledge of rationals, irrationals, and infinities is what lets the infinite series converge and hence allows us to 'arrive' at a particular destination - or number)
thanks!
John, I have a question another possible 'reality' for infinities.
Why not count motion? Bear with me...
If we don't have infinities, then we don't have irrational numbers and Xeno's paradox keeps us from ever arriving at our destination.
I realize there are multiple ... lets say routes to dealing with the resolution to Xeno's paradox but in one sense why not say that its the irrational numbers that let us arrive at a destination? - whether that destination be arriving at 1 from 0 or arriving at the finish line of a race.
I say this because since there is an infinite number of rational numbers between any two rational numbers we could have the paradox play out anytime we want to 'jump' or move anywhere. we can't step from one rational to another because of the 'gap' in between them. But once we have irrationals there is a continuity to physical space that lets us 'move', or at least prove that we did. If we were simply drawing a line with a pencil on a paper we couldn't get from A to B with only the rationals, but once we have the irrationals we can draw the line continuously. ... ok, a bit weird maybe, but could you under any stretch of the imagination by this is a 'physical' thing? (I'm also implying that this sort of knowledge of rationals, irrationals, and infinities is what lets the infinite series converge and hence allows us to 'arrive' at a particular destination - or number)
thanks!