The Root Cause of the Banach-Tarski Paradox

Page last updated 11 Jan 2026

The Banach–Tarski paradox Stefan Banach and Alfred Tarski, “PDF Sur la décomposition des ensembles de points en parties respectivement congruentes”, Fundamenta Mathematicae 6.1, 1924, pp.244-277.
There is an English translation: Andrew McFarland, Joanna McFarland, and James T. Smith, “On Decomposition of Point Sets into Respectively Congruent Parts (1924)” from the book Alfred Tarski: Early Work in Poland—Geometry and Teaching, Springer New York, 2014. pp.93-123, https://doi.org/10.1007/978-1-4939-1474-6_6. was devised by Stefan Banach and Alfred Tarski in 1924 using Zermelo-Fraenkel set theory. It claims that any solid sphere can be split into a finite number of disjoint sets of points (sets that have no points in common) such that those disjoint sets can be put back together in a different way to generate two spheres which are identical to the original sphere.

Stefan Banach We can note that the same argument leads to the conclusion that a single sphere of a given volume is equivalent to any finite number of spheres, each of the same volume as the original single sphere. You might think that sounds like a load of balls, and you would be right.

But almost all mathematicians whose principal activity revolves around Zermelo-Fraenkel set theory try to argue that that is not contradictory, that it only seems that way. For example, the the Wikipedia page Banach-Tarski pushes the line that it only seems to be a contradiction to the uninitiated, asserting that it “is not false or self-contradictory.”

Alfred Tarski But for you to accept that conclusion, you would also have to accept that it’s perfectly okay to have a mathematical system where you can assert that 1 = 2 when it suits you to do so, and at other times when it suits you to do so, you can assert that 1 ≠ 2 . This follows because if a mathematical theory asserts that the volume of one sphere is equivalent to the volume of two spheres, then it is also asserting that the radius of a sphere of unit radius is equivalent to the radius of a sphere of radius of two units, Since one sphere is equivalent to two spheres, then in the same way those two spheres are equivalent to four spheres, and those four spheres are equivalent to eight spheres, and since the volume of a sphere is proportional to the radius cubed, the volume of a sphere of radius 1 is the same as the volume of a sphere of radius 2, and hence the radii 1 and 2 are equivalent. and that is asserting that one unit is equivalent to two units, that is, that 1 = 2, and by extension, that one unit is equivalent to any finite number of units.

But here’s the crucial fact that determines that the Banach–Tarski result is a mathematical contradiction: in any mathematical system, once an equivalence is established between two entities, that equivalence applies everywhere in that system. To claim otherwise - that 1 is only equal to 1 when it suits but it  can be equal to anything else you choose at other times when it suits - is not mathematics, it’s absurdity. But many mathematicians appear to be delighted rather than appalled by its contradictory nature, claiming that it has a deep significance.

Background of the Contradiction

If you search through the available information on the Banach-Tarski paradox, you will see that it is universally accepted that the Banach-Tarski result cannot be achieved without the Axiom of Choice. Some people consider that the Banach-Tarski result is contradictory, and believe that the Axiom of Choice is the underlying cause of the contradiction. However, while the Axiom of Choice is required in the Banach-Tarski argument, in fact the principal source of the contradiction lies elsewhere.

Given that the Banach-Tarski result is a contradiction, we should analyze the argument carefully to see how that result is obtained. For example, we know that we cannot treat the infinite in the same way as we treat natural numbers - we don’t assume that we can treat the infinite symbol ∞ in the same way as natural numbers; we know that we cannot assume that since 2 × ∞ = ∞ then 2 = 1. And because we have learned that we can’t treat infinity in the same way as a natural number, we have analyzed why this is the case, and based on that analysis we have devised rules for the manipulation of ∞ in mathematical equations to prevent the improper use of ∞ that would lead to contradictions.

The first thing to observe is that the Banach-Tarski result depends on shifting points from one position to another. So, let’s first consider shifting points on the real number line. Suppose that we add 1 to all real numbers that are 0 or greater, so that we shift all those points on the real number line 1 unit to the right.

When we do this we don’t assume that the length of the line from 0 to ∞ is the same length as the line from 1 to ∞; they are both infinite lengths and hence, by our rules, they are not lengths to which our standard rules for comparison of finite lengths apply.

Now let’s move on to another example, this time involving a circle. First note that we can define a one-to-one correspondence between the points on the real number line and the points on a semi-circle, even though the line is infinitely long and the semi-circle is a closed line of length π. We can illustrate this as:

You can see that given any straight line going from the center of the partial semi-circle, through any point of the partial semi-circle, that line will intersect the real number line at some point. And from this we have a correspondence of all points from 0 to 1 on the real number line to the points of a semi-circle, excluding the points A and B (a straight line from the center going directly upwards is parallel to the fixed straight line and will never intersect it).

Now suppose we re-consider our previous shifting of points on the real number line from 0 to ∞ to 1 to ∞. By the correspondence to the top quarter-circle we have rotated all the points of the quarter-circle into a shorter arc between 1 and A. And although there is a one-to-one correspondence between the points of the original quarter-circle and the shifted points between 1 and A, that does not imply that the shorter arc is the same length as the original quarter-circle.

The rule that we use in this case is the rule of congruence, which is that a one-to-one correspondence between the points of two different sets of lines/arcs does not imply that the lengths of the two lines/arcs are equal unless exactly the same shifting of points is applied to all points. This rule extends to areas and volumes, i.e: a one-to-one correspondence between the points of two different sets of volumes does not imply that the volumes are identical volumes unless exactly the same shifting of points is applied to all points.

Rotating points of a circle

Now consider a full circle. Take the rightmost point (which is 1, 0 in the Cartesian plane), and locate the point which is at an arc length of exactly 1 unit counter-clockwise from that point 1 to give us our second point 2. We define a set P with this point 2 as the first point in the set. Now locate a point which is at an arc length 1 counter-clockwise from our point 2; this is point 3 and we add this as the second point in the set. Continue repeating the location of further points that are at an arc length 1 counter-clockwise from the previous point and add all such points to the set P.

Here’s the interesting bit. The arc lengths of all the multiple rotations are all multiple of units, while the length of the circumference of the circle is 2π . Hence the points of the multiple rotations can never coincide with the starting point 1, and also can never coincide with each other. So it seems that we have defined, by infinitely many rotations, an infinite set of points P.

Now suppose that we can rotate this set of points P clockwise through an arc of a length 1. We have now changed our set P that did not include the point 1 into a set that does include that point. This follows since our set of points P is infinite, and so there was no “last” point in the infinitely may repetitions of the addition of points separated by arcs of length 1. Thus we have done something very similar to removing the points of the real number line from 0 to 1, and then shifting all points from 1 to ∞ left by one unit to give the real number line from 0 to ∞.

Of course we would say that our set of points P around the circle do not constitute an arc, since they are all separated from each other. But suppose that we could do the same thing for all points of the arc of length 1, then since we would have transformed a circle with part of it missing into a complete circle, couldn’t we then claim that the length of the circumference of a circle is the same as the length of the circumference of a circle with part of it missing?

Clearly that would be absurd. The conventional response has been that since we cannot actually take an arc of length 1, and manipulate it in the same way as our set of points P, then then is nothing to worry about, and we do not need any additional rules such as we need for our infinity symbol ∞.

But what if someone could find a way around that problem - that while you cannot take an arc of length 1, and manipulate it in the same way as our set of points P, maybe there could be a more complex way of achieving some sort of equivalent result? Wouldn’t it be better to consider that that might happen, and analyze whether we should have an additional rule to prevent any contradiction arising?

The Hausdorff Paradox

Well, someone did find a way to generate a result similar to our hypothetical rotation of part of a circle - that is exactly what happened with the Banach-Tarski result. However, Banach and Tarski weren’t actually the first to find a way to achieve a result similar to our hypothetical rotation of part of a circle - the argument of Banach-Tarski’s paper relies completely on an earlier result by Felix Hausdorff, “By the time you receive these lines, we three have resolved the problem in another way – in the way that you have constantly tried to dissuade us from … What has happened against the Jews in recent months arouses well-founded fear that we will no longer be allowed to experience a bearable situation”
These are the opening words of a letter by Hausdorff, a German Jew, after he had received orders to report to a concentration camp in 1942. Hausdorff, who was then 73 years old, preferred to end his own life in his own way, by overdosing with sleeping pills along with his wife and her sister. as the Banach-Tarski paper itself admits. That result is commonly known as the Hausdorff Paradox, and can be seen in a paper by Hausdorff published in 1914. Felix Hausdorff, “Bemerkung über den Inhalt von Punktmengen”, Mathematische Annalen 75.3, 1914, pp.428-433.
Online English translation at A Remark on the Measure of Point Sets. The paper is in German, but you can read an English translation of Hausdorff ’s paper at A Remark on the Measure of Point Sets.

Felix Hausdorff While the Banach-Tarski claims to show that one solid sphere is equivalent to two solid spheres, Hausdorff ’s argument gives the result that one hollow sphere is equivalent to two hollow spheres, that is, he only considers the surface of a sphere, and comes up with the result that the surface of a sphere is equivalent to two such surfaces. The Banach-Tarski result follows inevitably from Hausdorff ’s argument. That means that we should first carefully examine Hausdorff ’s argument to see how he obtains his result.

Hausdorff ’s argument is based around the notion of multiple rotations of a sphere about two different axes, and he defines two specific rotations that he refers to as φ and ψ. The rotation φ is a rotation of 180° (half a complete rotation) and the rotation ψ is 120° (one third of a complete rotation). Hence, if two rotations of φ are performed successively, the sphere is back at its original position, and similarly, if three rotations of ψ are performed successively, the sphere is back at its original position. A crucial claim is that every rotation of any set of points always results in a set that is congruent to the original set of points, which means that the distance between any two points is always preserved in any rotation.

It is important to note that to facilitate matters Hausdorff considers that there are three fundamental rotations, which are φ, ψ and ψ². For convenience in the following, the word “rotation” always means one of these φ, ψ and ψ², while the term “m‑rotation” will be used to mean a sequence of multiple rotations (other than the fundamental rotation ψ²).

The interesting part about these m‑rotations is that one can choose the angle between the two different axes to be such that there there will be infinitely many different m‑rotations, so that while some m‑rotations will be repeats of a shorter sequence of rotations, there are always infinitely many m‑rotations that never repeat, just like the case of points on a circle that was discussed above.

Hausdorff ’s first trick is to separate the set of all the infinitely many m‑rotations into three disjoint separate subsets A, B, and C by a complicated set of rules. Disjoint sets are sets that have no elements in common, so in this case, none of the three sets have any m‑rotations in common. We don’t need to go into all the details of the rules and how they operate, it’s quite involved and takes some time to follow it completely. If you want to delve into the details, you can read my English translation of Hausdorff ’s paper. An important aspect of these rules is that they are recursive; each m‑rotation is defined in terms of previous shorter m‑rotations.

As such the set of all m‑rotations is denumerable, that is, they can be set in order and labeled as 1, 2, 3, etc. Similarly, each subset of m‑rotations A, B, and C is also denumerable. So we can we can refer to the set of all m‑rotations in A by a₁, a₂, a₃, …, a_n and similarly b₁, b₂, b₃, …, b_n for the m‑rotations in B and c₁, c₂, c₃, …, c_n for the m‑rotations in C. The rules for assigning m‑rotations to A, B or C are as follows:

a₁ is 1, b₁ is ψ and b₂ is φ, and the rules from these initial m‑rotations are: In Hausdorff ’s paper the rules are written as:
(i) If ψ_n belongs to A, B, C, then ψ_nφ belongs to B, A, A.
(ii) If φ_n belongs to A, B, C, then φ_nψ belongs to B, C, A, and φ_nψ² belongs to C, A, B.

1. If the last rotation of a_n is ψ, apply the rotation φ to give a_nφ which is in B.
2. If the last rotation of a_n is φ, apply the rotation ψ to give a_nψ which is in B, and apply the rotation ψ² to give a_nψ² which is in C
3. If the last rotation of b_n is ψ, apply the rotation φ to give b_nφ which is in A
4. If the last rotation of b_n is φ, apply the rotation ψ to give b_nψ which is in C, and apply the rotation ψ² to give b_nψ² which is in A
5. If the last rotation of c_n is ψ, apply the rotation φ to give c_nφ which is in A
6. If the last rotation of c_n is φ, apply the rotation ψ to give c_nψ which is in A, and apply the rotation ψ² to give c_nψ² which is in B

Having defined these three sets A, B, and C, Hausdorff ’s second trick is to take the the set A and from that set he defines three new sets of m‑rotations, as follows:

• He defines a set Aψ by applying the rotation ψ to every a_n to give a new m‑rotation that we can  refer to as a_nψ.
• He defines a set Aψ² by applying the rotation ψ² to every a_n to give a new m‑rotation that we can  refer to as a_nψ².
• He defines a set Aφ by applying the rotation φ to every a_n to give a new m‑rotation that we can  refer to as a_nφ.

Having defined these three additional sets Aψ, Aψ², and Aφ, Hausdorff ’s result is as follows:

• The set Aψ is the set B and A and B are congruent, If two point sets are congruent, that means that the distances between any two corresponding points is preserved by the rotation of one set into the other. The definition of congruence in the Banach-Tarski paper is:
  Point sets A and B are congruent,
  A ≅ B,
  if there exists a function φ that transforms A to B bijectively and satisfies this condition: for two arbitrary points α₁ and α₂ of the set A
  ρ(α₁, α₂) = ρ(φ(α₁), φ(α₂))
  where ρ(a, b) is the distance between the points a and b.
• The set Aψ² is the set C, and A and C are congruent,
• The set Aφ is the set B ∪ C, and A and B ∪ C are congruent.

And there we have it: the result that leads to the conclusion that 1 = 2.

The Contradiction

The contradiction arises from Hausdorff ’s second trick, which was to apply another rotation to the set A, and this assumes that the set A can be treated as though the selection of m‑rotations for the set A is already complete. But of course, there is no termination of the generation of new m‑rotations for the set A. You can observe that the definition of the set A necessarily involves infinitely many recursive definitions where its m‑rotations are determined by prior m‑rotations of the other sets B and C; and the definition of the set B necessarily involves infinitely many recursive definitions where its m‑rotations are determined by prior m‑rotations of the other sets A and C; and the definition of the set C necessarily involves infinitely many recursive definitions where its m‑rotations are determined by prior m‑rotations of the other sets A and B. Hence the definition of each set is an unending process which is never terminates, and the sets cannot be complete, since A cannot be complete until B and C are complete, and B cannot be complete until A and C are complete, and C cannot be complete until A and B are complete. Note, however, that the intertwined definitions of the sets A, B, and C only serves to highlight the non-terminating nature of the generation of new m‑rotations. The set of all m‑rotations also cannot terminate, and can never be complete.

We have a similar scenario to our earlier Rotating points of a circle scenario, where there is a definition of infinitely many points of a circle and the generation of further points is never terminated, but as more and more points are generated, individual points of the recursion become closer and closer with no lower limit to the distance between the points. The same applies to Hausdorff ’s sets, the generation of points on the surface of the sphere continues without limit, and the points become closer and closer together with no lower limit to the distance between them.

The limit state of the infinite recursions is that the points merge into each other, so for our points of a circle this results in the limit state being the entire circle. Similarly, for the generation of points of Hausdorff ’s sets the limit of each of Hausdorff ’s sets is the entire surface of the sphere - and so, if there are 3 such sets, you might then conclude that the surface of one sphere is the same as the surfaces of two spheres, exactly the same conclusion as Hausdorff ’s.

Of course, that conclusion is incorrect since those three limit sets are not disjoint but are simply different names for the same set, but the similarity to Hausdorff ’s result is most certainly not coincidental. Hausdorff ’s contradiction arises from the notion that all definitions of sets define sets that can always be considered as “complete” sets, rather than the definition of a set being simply a definition that has certain properties. See also contradictions that arise from the notion of “complete” sets in The Balls in the Urn Paradox and The Platonist Rod Paradox. This notion of “complete” sets prohibits set theorists from suggesting that Hausdorff ’s manipulations of certain sets might result in a contradiction as we have just done; since Hausdorff ’s argument follows the rules of Zermelo-Fraenkel set theory, then set theorists are obliged to accept it as a “correct” set theoretical result. This rigid adherence to that notion has led mathematicians, by way of Hausdorff, arriving at the result that 1 = 2 and then, rather than questioning what has led to that contradiction, they hunker down and claim that it will stand forever as an unassailably “true” mathematical theorem for all time.

This is the inevitable consequence of claiming that Zermelo-Fraenkel set theory is the final mathematical Theory of Everything, set in stone for eternity: set theorists have to accept every result that can be generated by that theory, and they have to follow that path wherever it leads, and so they have to claim that contradictions like the Hausdorff Paradox and the Banach-Tarski Paradox are not contradictions. See also Is Set Theory the Root of all Mathematics?

But no-one is obliged to follow the ideology of conventional set theory; we can perceive that mathematics is not set in stone forever, rather it is like the sciences, it evolves as new discoveries are made and new ideas come to the fore. And we acknowledge that the rules that mathematicians use should also change in response to such new ideas and new discoveries. It is interesting to note that Hausdorff ’s paper clearly considers the result to be contradictory, that Hausdorff himself believed that there was a mathematical problem that led to the contradiction and which needs to be addressed to prevent such contradictions.

In the case of the Hausdorff Paradox and the Banach-Tarski Paradox, there are options for creating a rule to prevent the generation of contradictions by the injudicious use of definitions that involve infinite recursion. We should take into account that some sets can be defined without using recursion, but whose elements can be enumerated, and the enumeration involves recursion; we do not want to include such sets in our rule. So the essence of the rule might be that if the definition of the elements of a set necessarily requires infinite recursion, then the set membership is that of the limit state of the recursion.

While Hausdorff ’s contradiction arises from the assumption of “complete” sets, the Banach-Tarski argument uses exactly the same underlying basis. It’s rather ironic that, far from it being a “true” theorem and only seemingly a paradox, is in fact a clear demonstration of why treating all infinite sets as though they are “complete” sets without taking into account whether their definition involves a non-terminating process of set inclusion might lead to contradictions. And while it is the case that the Axiom of Choice is involved in the generation of the result, the contradictory results of the Hausdorff and Banach-Tarski paradoxes are the direct result of the assumption that all infinite sets can be treated as if they are complete sets.

A short history behind the paradox

In 1905 Giuseppe Vitali showed that one could define certain sets for which Lebesgue’s theory of measure resulted in a contradiction. Giuseppe Vitali, “Sul problema della misura dei Gruppi di punti di una retta: Nota”, Tip. Gamberini e Parmeggiani, 1905.
Online English translation at On the problem of measuring sets of points on a straight line.
From this he concluded that there could be no solution to the problem of trying to create a perfect theory of measure of all possible sets such that the addition of the measures of any two or more sets will always give a correct result without any contradictions.

Nine years later, as discussed above, it was Felix Hausdorff ’s turn to assert the same claim - that there could be no solution to the problem of trying to create a perfect theory of measure of all possible sets such that the addition of the measures of any two or more sets will always give a correct result without any contradictions. He argued that his result was a real contradiction but he believed that the contradiction of his result arose from an impossibility of devising a theory of measure that can deal with all possible arbitrary sets of points.

It was another 10 years later that Banach and Tarski published their result. These three results occurred in the early years of the 20^th century, a time when set theory was in state of flux after the discoveries that Cantor’s set theory could lead to serious contradictions such as Russell’s paradox. There was considerable debate about these results. It had been noticed that each of Vitali’s, Hausdorff ’s and Banach-Tarski’s results relied on the use of the Axiom of Choice, and this became the central theme of the debate as to how to mange those results. While initially there was a very strong reaction by some mathematicians against the acceptance of the Axiom of Choice, among them Henri Lebesgue, Rene Baire, and Emil Borel, For example, Emil Borel wrote (“Lecons sur la théorie des fonctions”, Gauthier-Villars, 1914.)
My translation: “The contradiction originates in the application… of Zermelo’s axiom of choice. The set A is homogeneous on the sphere; but it is simultaneously a half and a third of it;… The paradox arises from the fact that A is not defined, in the logical and precise sense of the word ‘defined’. If one disregards precision and logic, one is led to contradictions.”
and:
“… any argument where one supposes an arbitrary choice a non-denumerably infinite number of times is outside the domain of mathematics”
  See: Loren Graham and Jean‐Michel Kantor, “A comparison of two cultural approaches to mathematics: France and Russia, 1890–1930.” Isis 97.1 (2006): pp.56-74, https://doi.org/10.1086/501100. the tide eventually went the other way, and set theorists generally decided to accept the Axiom of Choice and “solved” the problems of Lebesgue measure theory by simply declaring that Lebesgue theory did not apply to such sets, that they are “immeasurable”. And today almost all set theorists accept the Axiom of Choice as well as managing to hold the belief that Hausdorff ’s and Banach-Tarski’s results are not contradictory. For more on belief systems see Belief systems versus rationality.

Attempts at Justification of the Contradiction

For an example of attempts to show that Banach-Tarski is not contradictory, take a look at Leonard M. Wapner’s book The Pea and the Sun. Wapner, Leonard M. “The Pea and the Sun: A mathematical paradox”. AK Peters/CRC Press, 2005. Wapner, like many others, pushes the line that since it is not physically possible to decompose an actual ball in the way that the Banach-Tarski argument does, then Banach-Tarski is not actually a contradiction. But there are hundreds of real world scenarios where it is physically impossible to do certain tasks - but that does not mean that we cannot apply mathematics to analyze the scenario. Furthermore, for example, if we using 3 + 4 = 7 as part of a purely mathematical calculation, does that mean that we cannot use it later to apply to 3 objects and 4 objects?

Wapner writes about the ideas of a certain El Naschie which, he claims support the idea that Banach-Tarski is a valid result. El Naschie, M. S., “On the Initial Singularity and the Banach-Tarski Theorem”, Chaos, Solitons and Fractals 5(7), 1995, pp.1391-1392.
M. El Naschie, “Banach-Tarski Theorem and Cantorian micro space-time”, Chaos, Solitons and Fractals Chaos 5(8), 1995, pp.1503-1508. El Naschie is a former editor of the journal “Chaos, Solitons & Fractals”. When El Naschie was an editor of the journal he published hundreds of his own articles in the journal, even managing 5 articles in one issue, Chaos, Solitons and Fractals 5(38), 2008 and he took a legal libel case against the journal Nature, which he lost. His articles become more extreme in later years; one can get a good idea of the nature of El Naschie’s claims by visiting the El Naschie website where it proclaims:

Announcement of the resolution of the mystery of the hypothetically missing dark energy of the universe, the meaning of Einstein’s cosmological constant and a revision of Einstein’s iconic equation: E = mc² to become E = ^mc²⁄₂₂ where the 22 may be interpreted as the compactified dimension of the bosonic string theory of strong interaction (26 – 4 = 22 where the 4 are Einstein’s space time dimension and the 26 are Veneziano’s space time dimension).

or by this:

The tying of a knot in the rope shortens the end to end length. This reduction can be used to measure the so called open thickness energy of knots. There is an incredible correlation between this energy and particle physics. M.S. El Naschie, “Fuzzy multi-instanton knots in the fabric of space–time and Dirac’s vacuum fluctuation”, Chaos, Solitons and Fractals, 5(38), 2008, pp.1260-1268.

For more details on El Naschie see the Rational Wiki page on Mohamed El Naschie

We may also note how on the one hand Wapner pleads that Banach-Tarski is valid since it is not  possible to physically perform it, on the other hand Wapner also pleads that Banach-Tarski actually may have a physical counterpart; he refers to a paper by Bruno Augenstein, who makes vague “analogies” between Banach-Tarski and the production of particles in particle collisions. Augenstein, B., “Hadron Physics and Transfinite Set Theory”, International Journal of Theoretical Physics 23(12), 1984, pp.1197-1205