The Banach-Tarski paradox is a theorem in set theory that states a solid ball can be divided into a finite number of pieces, which, when rearranged using only rigid motions (translations and rotations) and without changing their shape, can form two identical copies of the original ball, effectively doubling its volume from nothing. It seems paradoxical because it defies physical intuition, but the "pieces" are not solid chunks but infinitely complex, scattered sets of points, made possible by the controversial Axiom of Choice in set theory, and involve non-measurable sets, meaning they have no defined volume, so no physical volume is truly created or destroyed.
What is the Banach-Tarski paradox in simple terms?
Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox.
The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed.
Tarski's theorem means that the solution set of a quantified system of real algebraic equations and inequations is a semialgebraic set (Tarski 1951, Strzebonski 2000). Although Tarski proved that quantifier elimination was possible, his method was totally impractical (Davenport and Heintz 1988).
It's a mathematical theorem involving infinity that makes it possible, at least in principle, to turn one apple into two. That argument is called the Banach-Tarski paradox, after the mathematicians Stefan Banach and Alfred Tarski, who devised it in 1924.
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".
The conclusion is that while mathematics (resp. logic) undoubtedly is more exact than any other science, it is not 100% exact. We cannot be 100% sure that a mathematical theorem holds; we just have good reasons to believe it. As any other science, mathematics is based on belief that its results are correct.
Banach-Tarski Theorem: If X and Y are bounded subsets of R3 having nonempty interiors, then there exist a natural number n and partitions {Xj : 1 ≤ j ≤ n} and {Yj : 1 ≤ j ≤ n} of X and Y respectively (into n pieces each) such that Xj is congruent to Yj for all j.
The controversy was over how to interpret the words "choose" and "exists" in the axiom: If we follow the constructivists, and "exist" means "find," then the axiom is false, since we cannot find a choice function for the nonempty subsets of the reals.
The most widely accepted contemporary theories of truth are [i] the Correspondence Theory ; [ii] the Semantic Theory of Tarski and Davidson; [iii] the Deflationary Theory of Frege and Ramsey, [iv] the Coherence Theory , and [v] the Pragmatic Theory .
Once a century, a very special day comes along. That day is today — 9/16/25. Pythagorean Theorem Day is a special date where the numerical representation of the date aligns with the Pythagorean theorem. Specifically, for the date includes perfect squares, and their square roots form a Pythagorean triple.
There isn't one single "most famous" paradox, but top contenders include Zeno's Paradoxes (like Achilles and the Tortoise) questioning motion, Russell's Paradox shaking mathematics' foundations, the Liar Paradox ("This statement is false") challenging logic, and the Grandfather Paradox in time travel, with the Fermi Paradox (where are the aliens?) also very well-known in science.
Now we can understand why it took them 379 pages just to prove 1+1=2. It's because they did not only intend to prove mathematics logically, but they also intended to give meaning to numbers like “1” and “2” as well as to symbols such as “+” and “=”.
You can say "I love you" in math through number codes like 143 (I-love-you, counting letters) or 520 (a Chinese code), using mathematical constants like the Golden Ratio (φ ≈ 1.618), or by representing it with equations or graphical heart shapes on calculators. More complex expressions involve programming syntax or creative calculus concepts.
1. Archimedes Paradox. Rather than taking a concern on the volume of water displaced, we need only take into account the volume of water surrounding the object. Thus arises the paradox where no fluid actually needs to be displaced for Archimedes' principle to take effect.
Banach Tarski is prooved by picking a representative from an infinite set (orbit equivalence classes). This requires the axiom of choice. However banach tarski does not imply the axiom of choice, they are not equivalent.
Without the axiom of choice, there can be a vector space with no basis, and there can be a vector space with bases of different cardinalities. Without the axiom of choice, the real numbers can be a countable union of countable sets, yet still uncountable.
In short, ZFC's resolved the paradox by defining a set of axioms in which it is not necessarily the case that there is a set of objects satisfying some given property, unlike naive set theory in which any property defines a set of objects satisfying it.
The proof of Tarski's Theorem is elementary when F is order-continuous, in addition to monotone. X be the smallest point in X, and let {xn} be the sequence of F-iterates from x; so xn = F(xn−1) and x0 = x. Since F is monotone, {xn} is a monotone sequence, and thus converges to a point e.
Banach-Tarski states that a ball may be disassembled and reassembled to yield two copies of the same ball. This is considered a paradox because it is contrary to geometric intuition that one can double the volume of an object by only cutting it up into pieces and rearranging these pieces rigidly.
The 6174 number trick, also known as Kaprekar's Routine, involves taking any four-digit number (with at least two different digits), arranging its digits to form the largest and smallest possible numbers, and subtracting the smaller from the larger; this process, when repeated, always converges to the number 6174, known as Kaprekar's Constant.
