The notion of infinity, of something that has no end, is evasive because it escapes the possibility of our experience. Any attempt to apprehend it is therefore unsuccessful, but perhaps that is what attracts us from the infinite, that is not to be caught.
The origins of the idea of infinity may have to be sought in remote cultures. It can be supposed that this idea has sprung up in some minds when contemplating the horizon or the wide sky, it may even have been the basis of some religious thesis. But the beginning of the study of infinity as a concept is found in Ancient Greece.
Perhaps it was Zeno who gave the initial kick with his famous paradoxes about infinity. Zeno with his paradoxes was actually defending Parmenides' philosophy against his detractors. Parmenides held that the universe was unique, eternal and immobile. And this idea was opposed to the philosophy of Heraclitus,famous for saying that nobody ever steps in the same river twice. But in reality what Zeno sought with his paradoxes was to prove the impossibility of movement, for which he used paradoxes present in the idea of infinity.
In order to clarify the concept of infinity, it was Aristotle who introduced a fundamental distinction. He considered two types of infinity or two different ways of conceiving it: the potential and the current. The current is the infinity in act, that is to say that it is understood as a finished whole. On the other hand, the potential is that which is interpreted as an operation consisting of always adding one more element to the last term, that is, it is something that is never done.
The fundamental difference between one and another way of understanding the concept of infinity can then be seen. Although Aristotle denied the existence of the current infinity and accepted the potential, not all agreed and that fundamental distinction opened a debate that among the philosophers was framed by the nominalist positions (defender of the Aristotelian position with respect to the infinity) and realistic (which affirmed the reality of the current infinity).
The debate about infinity would surely have been infinite if it had remained philosophically. But mathematical science would take the reins of this matter.
At first, mathematics subscribed to the infinite potential, since they did not need the idea of present infinity to demonstrate their postulates. But at the end of the 19th century the mathematician Georg Cantor introduced the transfinite numbers that are numbers conceived in such a way that they contain in themselves infinite elements, that is to say that the transfinites are infinite.
Cantor's ideas were imposed, giving rise to the development of set theory, in which the infinite is studied as a set.