Aren't we crossing infinity? Couldn't there be infinitely small time units? When a second passes, aren't we passing infinite units of time? When we walk across a room, aren't we passing an infinite amount of small length units? So aren't we "somehow" built to leap infinity?
I know my question is kinda premature, I was studying calculus and we were doing assumptions in infinity, like "infinity+1 is still infinity" and such, this thought came across my mind and I felt like this is one place I could ask it :)
 A: The current theory is that there is a smallest unit of distance (the Planck length) of 1.6 x 10^-35 metres and a smallest unit of time (Planck time) 5.4 x 10^-44 seconds.  And that these can not be divided any further. So time does not pass in a smooth manner, but more like the ticking of a clock.  So in answer to your question, no, there is not an infinite number of time or length units.
This all works with the concepts of quantum mechanics, whereby the very small can be in position 1 or position 2, 3, 4, ...   but cannot be in position 1.5, 2.5, etc.
Here is a paragraph from Fermilab   So why is the Planck length thought to be the smallest possible length? The simple summary is that it is impossible, using the known laws of quantum mechanics and the known behavior of gravity, to determine a position to a precision smaller than the Planck length. Pay attention to that repeated word "known." If it turns out that at very small lengths, some other version of quantum mechanics manifests itself or the law of gravity differs from our current theory, the argument falls apart. Since our understanding of subatomic gravity is incomplete, we know that the statement that the Planck length is the smallest possible length is on shaky ground. Still, until a better theory of quantum gravity is devised, the Planck length is the best estimate we have for a minimum length.
