Paradoxes in Mathematics and Beyond

Can an all-powerful being create a stone so heavy that it cannot lift it?

The greatest Greek liar says: “I am lying.” Is he telling the truth, or is he lying?

Does a set exist that contains all sets?

You might wonder what kind of questions these are—but they are paradoxes. Most of us, when faced with such puzzles, might respond like Socrates: “I know that I know nothing.” Yet even that response is, in itself, a paradox.

A paradox is any statement that seems to contradict itself while still appearing completely logical. Many paradoxes are known to be logically invalid, yet they are valuable because they stimulate critical thinking (for example, Zeno’s paradox or the Grandfather paradox). Others have revealed flaws in definitions once thought to be rigorous, leading to a reshaping of the very axioms of mathematics and logic (such as Russell’s paradox and Cantor’s paradox).

Take the liar’s paradox: the liar’s defining trait is that he lies. So if he says “I am lying,” then the statement must be false. But negating it gives us “I am not lying,” which implies he is telling the truth—yet he is a liar, so he cannot be telling the truth. This well-known paradox shows how even simple assumptions about truth and logic can lead to logical catastrophe. What is the broader significance of such results? From time to time, philosophers argue that the liar paradox tells us something profound about philosophy: perhaps that the world is, in some sense, “incomplete,” and that the idea of an all-knowing being is impossible. This sense of incompleteness carries into mathematics as well and extends through the sciences.

Consider the barber paradox: imagine a group of barbers who shave only those men who do not shave themselves. Suppose there exists a barber in this group who does not shave himself; then, by definition, he must shave himself. Yet no barber in the set can shave himself. How is this connected to mathematics? The barber paradox is actually a version of Russell’s paradox (formulated by Bertrand Russell, one of the greatest mathematicians of the 20th century).

To understand why this matters, we must recall that sets are fundamental to mathematics. Most mathematical objects (numbers, relations, functions) are defined through sets. A set’s defining property is that it contains elements, known as its members. Two sets are equal if they have exactly the same elements. Famous examples include the sets of natural, integer, rational, irrational, real, and complex numbers.

Before Russell’s paradox was discovered, mathematicians used what is called naive set theory. It defined a set as any well-defined collection of distinct objects. But the problem was the vagueness of the term “well-defined.” The assumption that any property could define a set—without restrictions—led directly to paradoxes. A classic example is Russell’s paradox: there is no such set as “the set of all sets that are not members of themselves.”

Russell framed it this way: let R be the set of all sets that are not members of themselves. If R is not a member of itself, then by definition it must be a member of itself. But if it is a member of itself, then it cannot be a member of itself, because it only contains sets that are not members of themselves. The contradiction is inescapable. By formulating paradoxes like this, Russell and others highlighted the importance of carefully designed rules for constructing sets. This work laid the foundation for modern mathematics in the 20th century.

In popular culture, paradoxes often appear in time-travel plots (Back to the Future), self-fulfilling prophecies (Harry Potter), satire, or depictions of everyday absurdities (Catch-22). In everyday language, the word “paradox” is also used for ironic or unexpected statements, such as the paradox that “standing still is more tiring than walking.”

To better understand paradoxes, logic and set theory, students often seek special math tutoring in Croatia – 'instrukcije iz matematike'. Online learning platforms such as ematematika.hr, one of Europe’s first specialized platforms for online math tutoring, provide resources and lessons that make these complex concepts more approachable.

Paradoxes often arise when we use absolute concepts like “everything” or “nothing.” Consider the thought experiment of an unstoppable force colliding with an immovable object. Or the classic theological question: If God is omnipotent, can He create a stone so heavy that He cannot lift it?

Mathematicians like Russell, Cantor, and Gödel worked on building the very foundations of mathematics to avoid such contradictions. Mathematics, as the language of science, aims to describe the world as it truly is—and paradoxes remind us of both its power and its limits.

eMatematika