  In this paper, a number of traditional models related to the percolation
theory has been considered by means of new computational methodology that does
not use Cantor's ideas and describes infinite and infinitesimal numbers in
accordance with the principle `The part is less than the whole'. It gives a
possibility to work with finite, infinite, and infinitesimal quantities
numerically by using a new kind of a computer - the Infinity Computer -
introduced recently in by Ya.D. Sergeyev in a number of patents. The new
approach does not contradict Cantor. In contrast, it can be viewed as an
evolution of his deep ideas regarding the existence of different infinite
numbers in a more applied way. Site percolation and gradient percolation have
been studied by applying the new computational tools. It has been established
that in an infinite system the phase transition point is not really a point as
with respect of traditional approach. In light of new arithmetic it appears as
a critical interval, rather than a critical point. Depending on "microscope" we
use this interval could be regarded as finite, infinite and infinitesimal short
interval. Using new approach we observed that in vicinity of percolation
threshold we have many different infinite clusters instead of one infinite
cluster that appears in traditional consideration.
