Modern infinity comes in many unequal sizes
Infinity names whatever is boundless or endless, written with the symbol ∞ that John Wallis introduced in 1655. Greeks argued over its nature for centuries, and at the end of the 19th century Georg Cantor showed that infinite sets can come in many different sizes.
Anaximander (c. 610–c. 546 BC) may be the earliest Greek on record with the idea; his word apeiron means unbounded or indefinite. Aristotle separated potential infinity from actual infinity, which he rejected because of the paradoxes it seemed to produce. Some argue the Hellenistic Greeks had a horror of the infinite, noting that Euclid said only that primes outnumber any assigned multitude. In India the Jain text Surya Prajnapti (c. 4th–3rd century BCE) sorted numbers into enumerable, innumerable, and infinite, each split into three orders.
Zeno of Elea offered no theory of infinity, yet his Achilles and the Tortoise paradox exposed weak popular notions, and Bertrand Russell called such puzzles immeasurably subtle and profound. In 1821 Augustin-Louis Cauchy gave a satisfactory definition of a limit and summed the geometric series. Running at 10 metres per second against a tortoise at 0.1 with a 100-metre start, Achilles draws level after about 10.1 seconds.
In the 17th century Europeans began using infinite quantities systematically. Wallis used his symbol in De sectionibus conicis for area calculations with infinitesimal strips, and in 1656 wrote out infinite series and products ending in “&c.” Newton discussed equations with infinitely many terms in 1699, and Leibniz treated infinitesimals and infinite quantities as ideal entities obeying the law of continuity. Still, nobody could say whether infinity was a number at all.
Cantor’s work showed, for instance, that the points on a line outnumber the integers. Zermelo–Fraenkel set theory, the usual foundation of modern mathematics, includes an axiom of infinity guaranteeing that infinite sets exist, and such sets turn up even in combinatorics. In calculus ∞ marks an unbounded limit rather than a real number. Physics has not settled whether the universe is spatially infinite.
Source: Infinity
