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I know that Kronecker claimed it was God's doing, and that even prehistoric humans used some ways of counting. But I am curious where the idea of a sequence of numbers stretching out into infinity appears for the first time explicitly. I suppose another way to phrase it would be to ask who invented infinity. "Ancient cultures had various ideas about the nature of infinity" is what Wikipedia says.

Pythagoreans must have understood it already, and Aristotle even discusses the difference between potential and actual infinities, so it had to be before that. Maybe they borrowed it from Egyptians? Babylonians had a positional system which allows recording arbitrarily large numbers in principle. But is that enough? What gives me pause is the history of zero. Babylonians and Alexandrian astronomers were using it as a placeholder for centuries before the concept of zero as representing nothing was formed, and not by them. And if it happened to zero it could happen to infinity. By the way, how ironic that infinity was discovered before zero.

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Conifold
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    Would you prefer an answer based on integers or infinity? – HDE 226868 Nov 11 '14 at 22:31
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    @HDE226868 This particular question is about "infinity of counting". I tacitly assumed that it was the first one conceived, but now that I think about it it's possible that geometric infinity of extension or divisibility preceded it. I was hoping that some early document like Rhind or Plimpton had "and so on" in it or something like that hinting at realization of indefinite continuation of integers, but the answers suggest that probably not. – Conifold Nov 12 '14 at 19:17

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Very few (if any) mathematicians before Cantor thought of the SET of integers. Certainly for Euclid it was completely evident that the sequence of integers extends without limit. (He actually has a famous theorem that the sequence of PRIMES extends without limit). Who discovered this we will never know because very few mathematical sources before Euclid survived. Perhaps Pythagoreans, but maybe earlier. (What we know about Pythagoreans comes from much later secondary sources, Pythagoreans themselves were a secret society and did not publish their discoveries).

As I said, most mathematicians imagined infinite sets as POTENTIALLY infinite, that is without limit. To every integer you can add 1 and obtain a larger integer.

However, in theological literature, beginning from early Middle Age, we encounter ACTUAL infinity (infinite things, infinite sets). I think this begins in the Neoplatonic school, but Augustine (of Hyppo) certainly discusses (in V century) that in the City of God, and Cantor mentions him. These discussion continue in the medieval scholastic literature, but there is little mathematics or science in them.

The notion of actual infinity was revived by Cantor in his set theory, and nowadays this is a common language of mathematics.

Remark. Since the Hellenistic times, it is a popular opinion that the first mathematicians (Pythagoras, Thales) "learned something from Egyptians". Some modern authors tend to say that they learned everything from Egyptians. Serious research on the history of mathematics and astronomy does not confirm that. With enormous number of surviving texts, we know pretty much about ancient Egypt. There was NOTHING for people like Thales to learn there. Egyptian astronomy and mathematics was in very primitive state in comparison with contemporary Babylonian and Greek sciences.

Best source: O. Neugebauer. Exact sciences in antiquity.

Alexandre Eremenko
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    I thought it was more or less uncontroversial that Pythagoras studied with priests in Egypt before founding his school. I agree that mathematics in surviving papyri is not overwhelming, but they deal with practical calculations, and priests were secretive, so we might not have the full story. Something in Egypt impressed Greeks enough to single them out as the only non-"barbarians". – Conifold Nov 12 '14 at 19:25
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    Even that Pythagoras existed is controversial. Both Pythagoras and Thales (according to the much later accounts) indeed traveled to Egypt. However, as I said there was not much in Egypt that they could learn and bring home. – Alexandre Eremenko Nov 12 '14 at 20:14
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    Quote from "Precession and the Pyramid Astronomical Knowledge in Ancient Egypt" by Jim Fournier "It follows that the ancient Greeks should be taken at their word when they claim that their knowledge is of great antiquity and was derived from Egyptian sources. Indeed it is nothing if not bizarre that modern scholars of the Greek world should go to great lengths to dismiss such claims on the part of the authors of the primary texts themselves, to instead rely on the advice of modern Egyptologists that the ancient Egyptians had no such knowledge." – Michael A. Sherbon May 17 '15 at 19:50
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    The title and contents you quote suggests that this is a pseudo-scientific nonsense, like most of the modern trash written about "astronomical knowledge" in connection with pyramids. – Alexandre Eremenko May 18 '15 at 05:02
  • Where are the enormous number of surviving texts from Egypt? You are not talking about the four or five papyri with mathematical content, are you?
    • –  May 18 '15 at 12:16
  • The fact that the earth is round is controversial, too. This is, however, not exactly an argument you can use in scientific circles.
    • –  May 18 '15 at 12:17
  • Read the "enormous number" of Greek sources testifying to their knowledge from the Egyptians .... many were even Egyptian initiates and knew how their knowledge was recorded in the geometry of their temples and pyramids. Extend your range of historical reading and maybe learn something about hieroglyphics. – Michael A. Sherbon May 18 '15 at 19:19
  • To the ancient statements that Thales learned something in Egypt, one modern historian commented: What could Thales learn in Egypt, if the same source says that he surprised Egyptians by measuring the height of the pyramid? What, indeed?? – Alexandre Eremenko May 18 '15 at 19:48
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    Anyway, if the advanced Egyptian mathematics/astronomy existed but all traces of it are completely lost, there is no subject to talk about. – Alexandre Eremenko May 18 '15 at 19:52
  • Not really lost, the hieroglyphics and and geometry are carved in stone for those able to read them. The knowledge does seem to be lost for modern minds who have not honestly searched for it. For philosophers, Plato, the Egyptian initiate, revealed some of the essential arguments about the mathematics of the Egyptians in allegorical form; part of the secret oral tradition that he was not allowed to reveal was passed on to Kepler, Newton, and others who acknowledged their debt to the Egyptians. – Michael A. Sherbon May 18 '15 at 21:20
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    Modern scientists are able to read hieroglyphs since the early 19s century. There is nothing substantial written on mathematics and astronomy with hieroglyphs. (Unlike the rich heritage of Babylonian cuneiform tablets). – Alexandre Eremenko May 19 '15 at 04:16
  • Something substantial ... "The celestial alignments of the Giza structures have been empirically demonstrated. The vast literature on the sophisticated knowledge of the ancients cannot be ignored (see, for instance, Stecchini, 1971)." --Robert Schoch, who writes like a scholar and a scientist, unlike the disposition and ignorance of some of his detractors. – Michael A. Sherbon May 19 '15 at 14:51
  • Boyer credits John Wallis, an important influence on Newton, as the first to use the scholastics' actual infinity in mathematics. – Jonathan Cender Jul 23 '19 at 21:40