Axioms of Real Numbers and Set Theory

The Foundation of Mathematics

Mathematical analysis begins with a question that seems trivial: what is a number? We are used to working with numbers since childhood, but a precise definition of a real number required three centuries of effort from the best mathematicians—from Newton and Leibniz to Cantor and Dedekind.

Real numbers form the number line—a continuous continuum, in which between any two numbers, a third can always be found. This property, called density, distinguishes real numbers from rational ones. Rational numbers are also "infinitely many," but they do not fill the number line completely—there are "gaps" between them, which are filled by irrational numbers such as √2 or π.

Cantor’s Set Theory

Georg Cantor, in the 1870s, created set theory—the language on which all modern mathematics is written. A set is any collection of definite and well-distinguished objects. Cantor made a striking discovery: there exist infinities of different "sizes".

There are infinitely many natural numbers {1, 2, 3, ...}—this is countable infinity. Surprisingly, there are as many rational numbers as natural: a one-to-one correspondence can be constructed between them (Cantor’s diagonal argument). However, there are more real numbers—they are uncountably many. Cantor proved this with his famous diagonal method: suppose all real numbers in the interval [0,1] can be written in a list. Then construct a number that differs from the first number in the list in the first decimal place, from the second—in the second decimal place, and so on. The resulting number will not coincide with any number in the list—a contradiction.

Axioms of Real Numbers

Real numbers are defined by a system of axioms, divided into three groups.

Field axioms guarantee the operations of addition and multiplication: commutativity ($a+b = b+a$), associativity, existence of zero and unity elements, existence of inverses.

Order axioms allow comparing numbers: for any two numbers $a$ and $b$, exactly one of the three is true: $a < b$, $a = b$, or $a > b$.

Completeness axiom (or Archimedean axiom in the equivalent formulation)—the key property distinguishing real numbers from rationals. It states: if a set $A$ is bounded above, then it has an exact upper bound (supremum). Informally: the number line has no "gaps." It is precisely this axiom that guarantees that $\sqrt{2}$ exists as a real number.

Operations on Sets

The main operations on sets are union $A \cup B$ (all elements belonging to at least one), intersection $A \cap B$ (only common elements), difference $A \setminus B$ (elements of $A$ not belonging to $B$), and complement.

De Morgan’s laws connect these operations: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$. These laws have analogues in logic and are used everywhere—from mathematics to programming.

Countable and Uncountable Sets

A set is called countable if its elements can be numbered with natural numbers. Countable: natural, integer, rational numbers. Uncountable: real numbers, the segment [0,1], any interval of nonzero length.

This distinction is fundamental. When a computer stores a number in memory, it works with a finite approximation—a rational number with a finite number of digits. An infinity of real numbers, inaccessible for exact representation, yawns between any two "computer" numbers. This is the foundation of the theory of computational errors.

Practical Significance

Set theory and the axiomatic system of real numbers are not an academic abstraction. They underpin all of mathematical analysis, and therefore all modern physics, engineering, and computer science. Understanding that the number line is "without gaps" allows us to guarantee the existence of solutions to differential equations, roots of functions, extrema—everything on which mathematical modeling is built.

Question for reflection: Prove that between any two rational numbers $p/q$ and $r/s$ there exists an irrational number. Hint: consider the number $(p/q + r/s)/2 + \delta$, where $\delta$ is a small irrational.

Connection with Computer Science and Logic

Set theory is a programming language in a broad sense. Operations on sets have direct analogues in SQL (UNION, INTERSECT, EXCEPT), in Boolean algebra of logical circuits, and in type theory in programming languages. When a programmer writes "a set of keys" in a hash table or "a set of unique elements" in a standard library, they are embodying Cantor’s idea.

Russell's paradox (1901) shook mathematics: consider "the set of all sets not containing themselves." Does it contain itself? If yes—it should not. If no—it should. This paradox led to the creation of the axiomatic set theory of Zermelo–Fraenkel (ZFC)—the standard foundation of mathematics. In computer science, the analogue of this paradox is the Turing halting problem: it is impossible to write a program that determines whether an arbitrary program will finish.

The Cardinality of Sets and Cantor’s Diagonal Argument

Cantor proved that the cardinality of real numbers is strictly greater than that of natural numbers. Method of diagonalization: enumerate all the numbers in [0,1] in a list $x_1, x_2, x_3, ...$ Construct a number $y$, whose $n$-th decimal digit differs from the $n$-th digit of $x_n$. Then $y \neq x_n$ for all $n$—contradiction. Cardinality of natural numbers $\aleph_0$, cardinality of real numbers $\mathfrak{c} = 2^{\aleph_0}$. Continuum hypothesis: does there exist a cardinality strictly between $\aleph_0$ and $\mathfrak{c}$? Gödel (1938) and Cohen (1963) proved that this question is undecidable in the axiomatic system ZFC—one of the most striking results of mathematical logic of the 20th century.