09-07-2026, 11:14 PM
Summary
Georg Cantor’s 1874 paper “On a Property of the Collection of All Real Algebraic Numbers” is generally regarded as the article that launched modern set theory. Cantor first proved that the set of real algebraic numbers is countable: they can be arranged in a sequence $x_1,x_2,x_3,\ldots$ and therefore put into one-to-one correspondence with the positive integers. His method orders integer polynomials according to a measure of their degree and coefficients and then lists their real roots.
The paper's revolutionary step is Cantor's second theorem. Given any sequence of real numbers $x_1,x_2,x_3,\ldots$ and any interval $[a,b]$, Cantor constructs nested intervals and proves that there must exist a real number inside $[a,b]$ that is not contained in the sequence. Consequently, the real numbers cannot be enumerated by the natural numbers: $\mathbb{R}$ is uncountable. Importantly, this 1874 argument is not Cantor's later diagonal argument; it relies instead on nested intervals. Combining this result with the countability of the algebraic numbers also shows that every real interval contains infinitely many transcendental numbers.
The paper fundamentally changed the mathematical understanding of infinity by demonstrating that infinite collections need not all have the same size. Cantor later formalized the comparison of infinite cardinalities and developed ordinal and cardinal arithmetic, while the notions of countability and uncountability became central to analysis, topology, measure theory, mathematical logic, and the foundations of mathematics.
Key takeaways
ARTICLE
Georg Cantor’s 1874 paper “On a Property of the Collection of All Real Algebraic Numbers” is generally regarded as the article that launched modern set theory. Cantor first proved that the set of real algebraic numbers is countable: they can be arranged in a sequence $x_1,x_2,x_3,\ldots$ and therefore put into one-to-one correspondence with the positive integers. His method orders integer polynomials according to a measure of their degree and coefficients and then lists their real roots.
The paper's revolutionary step is Cantor's second theorem. Given any sequence of real numbers $x_1,x_2,x_3,\ldots$ and any interval $[a,b]$, Cantor constructs nested intervals and proves that there must exist a real number inside $[a,b]$ that is not contained in the sequence. Consequently, the real numbers cannot be enumerated by the natural numbers: $\mathbb{R}$ is uncountable. Importantly, this 1874 argument is not Cantor's later diagonal argument; it relies instead on nested intervals. Combining this result with the countability of the algebraic numbers also shows that every real interval contains infinitely many transcendental numbers.
The paper fundamentally changed the mathematical understanding of infinity by demonstrating that infinite collections need not all have the same size. Cantor later formalized the comparison of infinite cardinalities and developed ordinal and cardinal arithmetic, while the notions of countability and uncountability became central to analysis, topology, measure theory, mathematical logic, and the foundations of mathematics.
Key takeaways
- Cantor proved that the algebraic numbers are countably infinite.
- He proved that $\mathbb{R}$ is uncountable, establishing different sizes of infinity.
- His original 1874 proof used nested intervals, not the famous diagonal argument.
- The paper is widely considered the starting point of modern set theory.
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│ KONSTANTINOS MICHAILIDIS │
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