  In this article, some classical paradoxes of infinity such as Galileo's
paradox, Hilbert's paradox of the Grand Hotel, Thomson's lamp paradox, and the
rectangle paradox of Torricelli are considered. In addition, three paradoxes
regarding divergent series and a new paradox dealing with multiplication of
elements of an infinite set are also described. It is shown that the surprising
counting system of an Amazonian tribe, Pirah\~{a}, working with only three
numerals (one, two, many) can help us to change our perception of these
paradoxes. A recently introduced methodology allowing one to work with finite,
infinite, and infinitesimal numbers in a unique computational framework not
only theoretically but also numerically is briefly described. This methodology
is actively used nowadays in numerous applications in pure and applied
mathematics and computer science as well as in teaching. It is shown in the
article that this methodology also allows one to consider the paradoxes listed
above in a new constructive light.
