Set Theory

What is infinity, and can all of mathematics be built out of collections?

5 chapters4 min read5 turning points 1 open problem

Branched from: Mathematical Logic + Real Analysis
Branched into: Metamathematics
Figures: Georg Cantor, Bertrand Russell, Ernst Zermelo, Kurt Gödel, Paul Cohen

In brief

Set theory studies collections, or sets, and above all infinite ones. Its founding discovery is that infinities come in different sizes: there are exactly as many even numbers as whole numbers, but strictly more real numbers than either. Building on that, set theory became the common material from which all of modern mathematics is constructed, with numbers, functions and spaces all defined as sets.

Its history is one of crises. Naive reasoning about sets produced paradoxes, the axiom meant to repair them provoked a feud, and the most natural question about infinity, the continuum hypothesis, turned out to be impossible to settle from the standard axioms at all.

Key ideas

CardinalityEnters 1891: Two sets have the same size if their elements can be paired off one-to-one. By this measure the even numbers and the whole numbers are the same size, even though one is part of the other.
Countable and uncountableEnters 1891: A set is countable if it can be listed $1, 2, 3, \ldots$ . The rational numbers are countable; Cantor's diagonal argument shows the real numbers are not.
Power set and Cantor's theoremEnters 1891: The set of all subsets of any set is strictly larger than the set itself. So there is no largest infinity: every infinity is followed by a bigger one.
Axiom of choiceEnters 1904 – 1908: From any collection of non-empty sets, one element can be chosen from each, even when there is no rule for choosing. Indispensable in modern mathematics, and it implies strange results such as the Banach–Tarski paradox.
IndependenceEnters 1938 – 1963: A statement is independent of a set of axioms if neither it nor its negation can be proved from them. The continuum hypothesis is independent of the standard axioms of set theory, ZFC.

Chapter I

Sizes of Infinity

Set theory grew out of real analysis. Studying where a Fourier series could misbehave, Georg Cantor was led in the 1870s to think about infinite sets of points as objects in their own right, and to ask how big they are. His answer was to compare sets by pairing their elements. The even numbers pair perfectly with all whole numbers ( $1 \leftrightarrow 2$ , $2 \leftrightarrow 4$ , …), so they are the same size, even though one is a part of the other. So are the fractions, which can be listed cleverly.

But the real numbers cannot. In 1891 Cantor gave the diagonal argument. Take any list of real numbers and build a new one that differs from the first in its first digit, from the second in its second, and so on. It cannot be on the list. There are strictly more real numbers than whole numbers, and the same idea shows there is no largest infinity at all. Leopold Kronecker is said to have called Cantor a "corrupter of youth", while Hilbert declared that "no one shall expel us from the paradise that Cantor has created."

Chapter II

Paradise Lost

Cantor asked the natural next question: is there any infinity between the whole numbers and the real numbers? He conjectured not, the continuum hypothesis, and could not prove it.

Worse was coming. Frege and others had assumed that any property defines a set, the set of things having it. In 1902 Bertrand Russell wrote to Frege about the set of all sets that do not contain themselves. If it contains itself, it does not, and if it does not, it does. Frege, whose second volume was at the printer, added an appendix admitting that the foundation of his work had collapsed. Ernst Zermelo had found the same paradox in Göttingen and not published it.

Chapter III

Axioms and a Feud

Zermelo's response was to replace "any property defines a set" with careful axioms saying which sets exist. In 1904 he made explicit a principle mathematicians had used without noticing: the axiom of choice, that one can always pick an element from each of any collection of non-empty sets. The French analysts Borel, Baire and Lebesgue attacked it for asserting the existence of objects nobody could construct, and when Banach and Tarski later used it to cut a ball into finitely many pieces and reassemble them into two balls of the same size, the objection seemed vindicated. The axiom stayed because too much of mathematics needs it. With Abraham Fraenkel's refinements, Zermelo's system became ZFC, the standard foundation of mathematics today.

Chapter IV

A Closer Look: Listing the Fractions, and Failing to List the Reals

It seems obvious that there are more fractions than whole numbers: between any two whole numbers lie infinitely many fractions. Cantor showed it is false. Arrange all positive fractions $\frac{p}{q}$ in a grid, numerator along the rows and denominator along the columns, and walk it diagonal by diagonal:

\tfrac11,\ \tfrac12,\ \tfrac21,\ \tfrac31,\ \tfrac22,\ \tfrac13,\ \tfrac14,\ \tfrac23,\ \tfrac32,\ \tfrac41,\ \ldots

Skip any fraction already seen in lower terms (like $\frac22 = \frac11$ ), and every positive fraction receives a position in one list. So the fractions are countable: exactly as numerous as the whole numbers.

The real numbers are different. Suppose someone claims to list every infinite sequence of 0s and 1s (each real number between 0 and 1 has a binary expansion):

position	sequence
1	0 1 1 0 1 …
2	1 1 0 0 1 …
3	0 0 0 1 1 …
4	1 0 1 1 0 …

Read down the diagonal (0, 1, 0, 1, …) and flip every digit to get $1, 0, 1, 0, \ldots$ . This new sequence differs from the first entry in position 1, from the second in position 2, from the $n$ th in position $n$ . It is on no line of the list. Every attempted list misses something, so the reals are uncountable, a strictly larger infinity.

The same diagonal trick, turned on sets and their subsets, proves there is no largest infinity. Turned on formulas and proofs, it gives Gödel's incompleteness theorem. Turned on programs, it gives the unsolvability of the halting problem. Few arguments in mathematics have travelled further.

Chapter V

A Question Without an Answer

Hilbert put the continuum hypothesis first on his 1900 list of problems. The answer came in two halves. In 1938 Kurt Gödel showed it cannot be disproved from ZFC. In 1963 Paul Cohen, an analyst new to logic, invented forcing, a method for building new models of set theory, and showed it cannot be proved either. The most natural question about infinity is independent of the axioms, just as the parallel postulate was independent of Euclid's others. Whether it nonetheless has a true answer is argued to this day. Meanwhile the effort to secure mathematics from paradox led to metamathematics, and to Gödel's discovery of limits no foundation can escape.

Turning points5

1878CONJECTURE
Cantor's continuum hypothesis
Having shown that the real numbers form a larger infinity than the whole numbers, Georg Cantor conjectures that there is no infinity in between: every infinite set of real numbers is either countable or as large as all the reals. He spent much of the rest of his life trying to prove it. Hilbert made it the first of his 23 problems in 1900.
Georg Cantor
› Sources (1)
- Cantor, G. (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik 84: 242–258.
1891PROOF
The diagonal argument
Cantor gives a two-line proof that the real numbers cannot be listed. Given any list, build a number that differs from the $n$ th entry in its $n$ th digit, and it is missing from the list. The same idea shows every set has more subsets than elements. Diagonalisation would later power Gödel's and Turing's theorems.
Georg Cantor
› Sources (2)
- Cantor, G. (1891). Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung 1: 75–78.
- Dauben, J. W. (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press.
1901 – 1903CRISISContested
Russell's paradox
Consider the set of all sets that do not contain themselves. Does it contain itself? Either answer contradicts itself. Bertrand Russell sends the paradox to Gottlob Frege in 1902, just as Frege's life work on the foundations of arithmetic is going to press. Frege's system, and naive set theory with it, is inconsistent.
What is disputed
Ernst Zermelo appears to have found the same paradox around 1899–1901 and communicated it to colleagues in Göttingen, including Hilbert, without publishing. Russell published it, and the paradox bears his name. Related paradoxes of the largest cardinal and ordinal (Cantor, Burali-Forti) were known slightly earlier.
Bertrand Russell, Ernst Zermelo
› Sources (1)
- van Heijenoort, J. (ed.) (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press.
1904 – 1908REFORMULATIONContested
Zermelo axiomatises set theory
To prove that every set can be well-ordered, Ernst Zermelo makes explicit the axiom of choice (1904). In 1908 he publishes axioms for set theory designed to block the paradoxes. With Abraham Fraenkel's additions in 1922 they became ZFC, the standard foundation of mathematics.
What is disputed
The axiom of choice set off one of the fiercest controversies in modern mathematics. Borel, Baire and Lebesgue rejected it as asserting the existence of objects no one could define, while Hadamard defended it. Its later consequences, such as the Banach–Tarski paradox (1924), in which a ball is cut into finitely many pieces and reassembled into two balls of the same size, kept doubts alive. It is now accepted by most mathematicians, though some constructive schools still reject it.
Ernst Zermelo
› Sources (2)
- Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65: 261–281.
- Moore, G. H. (1982). Zermelo's Axiom of Choice: Its Origins, Development, and Influence. Springer.
1938 – 1963PROOF
The continuum hypothesis is independent
Kurt Gödel shows in 1938 that the continuum hypothesis cannot be disproved from the axioms of set theory. In 1963 Paul Cohen, inventing the method of "forcing", shows it cannot be proved either. Hilbert's first problem has an answer no one expected: the standard axioms cannot settle it. Cohen received the Fields Medal in 1966.
Kurt Gödel, Paul Cohen
› Sources (2)
- Gödel, K. (1938). The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proceedings of the National Academy of Sciences 24(12): 556–557.
- Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences 50(6): 1143–1148.

Open problems

Where the map runs out

Open

Is the continuum hypothesis true?

Independent of ZFC. Whether new axioms should, or can, settle it is disputed as of 2026.

Independence means ZFC cannot decide the question. Many set theorists think it still has a definite answer, to be found by adopting new, well-motivated axioms. Others argue there is no single universe of sets, only many, some where the hypothesis holds and some where it fails.

Why it is hard: Candidate new axioms, such as large-cardinal axioms, settle many independent questions but provably leave the continuum hypothesis open. Hugh Woodin has pursued a programme ("Ultimate L") in which it would be true, having earlier argued for its failure. Joel David Hamkins and others defend a "multiverse" view in which the question has no single answer. The disagreement is partly mathematical and partly philosophical.
What resolving it unlocks: It would shape what mathematicians take the foundations of their subject to be: one definite world of sets, or many.

› Sources (2)

Woodin, W. H. (2001). The continuum hypothesis, Part I. Notices of the AMS 48(6): 567–576.
Hamkins, J. D. (2012). The set-theoretic multiverse. Review of Symbolic Logic 5(3): 416–449.