Given a completely empty space, it is accepted from intuition and modern physics that that space is infinitely divisible. That is, there is no smallest part of space—we can always imagine smaller. The notion of an instant or point is commonly used and accepted in physics (think ‘instantaneous velocity’). Note that a mathematical point has no volume, area and is the ‘physical’ representation of the infinitesimal (0.0...1).
It seems so basic, when considering a truly empty space void of all matter, that it is divisible ad infinitum. I am inclined to believe that my following logic must be flawed due to this so intuitive and evident notion. However, I will show the problems with this ‘undeniable’ assumption:
1. Matter exists.
2. Matter occupies space.
3. Space is infinitely divisible.
4. It follows that whatever occupies this infinitely divisible space must in turn be infinitely divisible. Thus, matter is infinitely divisible.
5. Dividing matter indefinitely will lead to mathematical points that do not exist in space (that is, they do no occupy space).
6. Since the parts of matter do not exist, the whole of matter does not exist.
Also, an even more problematic situation occurs when we consider
Zeno’s Paradoxes which I defend are logically valid and unsolved by calculus that assumes infinite series can sum to a limit. That is, calculus assumes the infinitesimal is equal to 0, yet then allows for us to divide by it when considering limits (ex. Lim x->0 5/x equals infinity) or, in physics, instantaneous velocity (which is defined by the limit as time->0 in the equation v=d/t... if it was zero we couldn’t calculate instantaneous velocity).
However, in order to amend these problems (i.e. continue believing that motion is possible and that matter exists), we have a few options:
1. Space (and time) is not infinitely divisible. Or,
2. Within infinitely divisible space and time lies matter that is bound by some unknown law or reason that allows it only to occupy certain portions or units. That is, matter can only exist and interact/move in certain smallest units of space. This implies that motion is not continuous but rather a vibration of miniature teleportations. Also, this means that despite space being infinitely divisible, the matter that resides in it is not. It is hard to understand just how this would work.
Please discuss whether space is infinitely divisible.