A tiny line segment holds exactly as many points as an endless line
Take a segment one centimetre long and a line stretching forever in both directions. Intuition says the line has more points. Set theory says they have exactly the same number. Georg Cantor's study of infinite collections produced surprises like this, and it ended up reshaping the foundations of mathematics itself.
A set is simply a collection of distinct things, its elements, which can be numbers, points, shapes, functions or even other sets. Oddly, the idea cannot be formally defined: it is a primitive notion, pinned down instead by axioms describing how collections should behave. One such rule, extensionality, says two sets are identical exactly when they contain the same elements. From that it follows that there is only one empty set, and that a set holding just the empty set is different from the empty set, since one has an element and the other has none.
Before the late 19th century, sets were barely studied and hardly told apart from sequences. Most mathematicians thought of infinity as potential, the outcome of a process that never ends, and avoided infinite collections. A line was not a set of points but a place where a point might be found.
Cantor, who lived from 1845 to 1918, changed that. He showed that the real number line holds a strictly bigger infinity than the natural numbers, even though both are endless. Then trouble arrived. Assuming a set of all sets exists leads to a contradiction known as Russell's paradox, which helped trigger a foundational crisis. Of the proposed fixes, Zermelo–Fraenkel set theory became the generally accepted foundation, although most everyday mathematics never needs its full strength.
Since the first half of the 20th century, every mathematical object has ultimately been defined in terms of sets. Spaces and algebraic structures are built from them, and old results have been restated in their language: Euclid's theorem becomes the statement that the set of primes is infinite. David Hilbert foresaw this spread, vowing that mathematicians would never be expelled from the paradise Cantor had built.
Source: Set (mathematics)
