The problem  here is that the walker  is required  to
　　   traverse  an infinite  series  of distances, which is
　　   impossible.  Since  time  is  discrete, in  order  to
　　   traverse  each of the distances  involved, the walker
　　   requires  at least one minim of time.  Therefore  the
　　   journey requires an infinite number of such minims of
　　   time, that  is, an infinite  duration, and  for  this
　　   reason it can never be completed.
　　　　  The paradox of Achilles  and the Tortoise assumes
　　   that space is discrete  and time continuous.  It goes
　　   as follows:

　　   The second is the so-called  Achilles, and it amounts
　　   to this, that in a race the quickest runner can never
　　   overtake  the slowest, since  the pursuer  must first
　　   reach the point whence  the pursued  started, so that
　　   the slower must always hold a lead.  This argument is
　　   the  same  in  principle  as that  which  depends  on
　　   bisection, though  it  differs  from  it in that  the
　　   spaces  with which  we successively  have to deal are
　　   not divided into halves.(5)

　　   In this  case, the difficulty  arises  from  the fact
　　   that there is an infinite  series of moments in which
　　   the tortoise is running. In each moment, the tortoise
　　   must traverse  at least one minim of space.  In order
　　   to overtake the tortoise, Achilles must traverse each
　　   spatial minim through which the tortoise  has passed.
　　   Therefore, Achilles  would have to travel an infinite
　　   distance  in order  to catch  the tortoise.  Like the
　　   Bisection  Paradox, this problem can be simply stated
　　   thus: one can never complete an infinite series.
　　　　  The  Arrow  Paradox, on the  other  hand, assumes
　　   both space and time to be continuous.

　　   The third is that already  given above, to the effect
　　   that  the  flying  arrow  is  at  rest, which  result
　　   follows from the assumption  that time is composed of
　　   moments: if  this  assumption  is  not  granted,  the
　　   conclusion will not follow.(6)

　　   Because  time  is infinitely  divisible, and  because
　　   moments  thus  have no duration, at any given  moment
　　   the arrow is standing  still in a space  equal to its
　　   length. Therefore, it is at every moment at rest, and
　　   thus it never moves.  Once again, the problem  can be
　　   simply   stated:  one   cannot   add   a  number   of
　　   dimensionless