That's the paradox. People think a larger size implies more points in a solid object (common sense) when such isn't the case. Just because I didn't say the points are dimensionless in my OP doesn't imply each point has a size, in fact it's assumed they don't individually have a size. It's proven by Cantor that both sets of points are the same in number (Aleph null) through one-to-one correspondence (using Cartesian coordinates). And there you have it.Arising_uk wrote: ↑Sat Apr 07, 2018 7:34 pm It doesn't matter if it's size or distance your points are DIMENSIONLESS and as such will have no relationship to size or distance.
The reason why you think you have a paradox that defies common-sense is that in your OP you are implicitly implying that points are not dimensionless and as such are in some way filling up the space so when you compare an inch line with an inch square common-sense would imply that there should be more points in the latter and if in reality points did work like this then they'd be dimensioned and it would be a paradox that there are the same amount of points in both but you justify your claim that there are by using these dimensionless points and by bloody definition these can have no relation to size or distance so the 'deduction' is pointless.
PhilX
It's not that I can't se them that I think they don't exist it's because they are a logical contradiction or at best a mathematical abstraction.