Os Limites da Intuição ao Lidar com o Infinito

Abstract

This work aims to elucidate the notion of “equal infinities” through the concept of bijection, which is a one-to-one correspondence between two sets of elements. To contextualize the theme, we return to a time before the creation of numbers, when shepherds would associate each animal in their flock with a stone, ensuring at the end of the day that no animal had been lost. This rudimentary method illustrates the fundamental principle of bijection: if it is possible to establish a one-to-one correspondence between two sets, then they have the same cardinality (number of elements). Building on this idea, we extend the discussion to infinite sets, presenting a counterintuitive example: the set P of even natural numbers and the set N of all natural numbers have the same cardinality. At first, this statement seems absurd, since P is a proper subset of N. However, by constructing an explicit bijection between the elements of P and N, we demonstrate that they indeed have the same cardinality. Through this and other examples, this study seeks to clarify the concept of cardinality, highlighting the peculiarities of infinite sets and challenging intuitive notions of size and quantity.