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Russell's paradox showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work.

Now, Banach–Tarski paradox is arises from a result that a ball can be decomposed into finite amount of pieces and the pieces can be used to built two identical copies of the decomposed ball. Banach-Tarski paradox is often treated as a "paradox", basicly meaning that, yes, it is counter intuitive but yet there is no problem - mathematics just occasionally is counter intuitive.

To be honest, I have never understood why Banach-Tarski is not a "real" paradox but not being expert of measure theory I chose to accept the common view.

Is there some high level explanation on how to tell a paradox from a "paradox"? What is it that makes a counter intuitive result to a "real mathematical paradox" that we should start worrying about?

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The point of Banach-Tarski is not just that one ball equals two balls - it is that the disassembly and assembly maps are rigid motions of Euclidean space, and hence there is no non-zero functional that is defined on all subsets and is invariant under the isometry group of Euclidean space. This only works in dimensions 3 and above; there is no B-T paradox IIRC in dimensions 1 or 2. –  Yemon Choi Sep 26 '12 at 7:57
@Yemon: I think you forgot finite additivity in your list of conditions. :-) Anyway, Terry Tao gave a good description of the difference between the dimensions on his blog: terrytao.wordpress.com/2009/01/08/… –  Willie Wong Sep 26 '12 at 8:44
@WillieWong (1) quite correct, oops (2) I knew that (non-)amenability was the main issue here but I couldn't recall the precise details in haste –  Yemon Choi Sep 26 '12 at 8:55
Note that in mathematics, and even in the current language, paradox should refer to the former acception you mentioned, that is, a counter-intuitive fact, something that is contrary to the common opinion, thus a priori not dangerous. For the latter notion, I think you mean antinomy, something that leads to a contradiction. –  Pietro Majer Sep 26 '12 at 9:43
Voting to close as "not a real question" (not even from a metamathematical or philosophical viewpoint) –  Qfwfq Sep 27 '12 at 8:05

5 Answers 5

up vote 8 down vote accepted

Many paradoxes are first expressed in a semi-formal way, for example "the least number not describable by fewer than eleven words". They are warning signs that lead us to further analysis and can be resolved in different ways:

  1. We can just get used to a "paradox" and accept it as "truth", e.g., there are infinite sets of different sizes, or there is a real function which is continuous at irrational arguments and discontinuous at rational arguments. There are famous paradoxes in philosophy which would not be considered paradoxes today, such as Zeno's paradox ("How can an infinite sum of positive numbers be finite? No movemement is possible!") and various arguments from Prime Cause ("How could we have an infinite descending chain of causality? God must exist!").

  2. We find the paradox unacceptable and so we need to change something. We might change rules of logic, definitions, or axioms, everything is up in the air.

A paradox which actually proves falsehood, or a statement as well as its negation, is more properly called an inconsistency. An inconsistency is something we can never get used to and so we have to change something. A milder form of paradox is one which does not prove falsehood but just something very counter-intuitive, in which case we have to decide whether to accept it, or admit that our attempt to bring something into the realm of mathematics worked in unexpected ways.

I think this question is about how to tell whether a given "paradox" is of the first or second kind. When should we just "get used" to a paradox and when should we "change things"? In the case of Russell paradox we had no choice but to change something. In the case of Banach-Tarski paradox there is a choice. The accepted view is that we should just get used to it, but there are interesting alterantives which force us to rethink the notion of space. Even though these alternative notions of space are far better suited for probability, measure and randomness than the classical approach, mathematicians are unlikely to adopt them widely out of sheer inertia and historical coincidence. But mathematicians do not like to admit that mathematics is a human activity, and as such subject to sociological and historical trends.

So I suppose my answer is this: when faced with an unacceptable counter-intuitive statement which offers several mathematical resolutions, the choice will be made through social interaction which has some mathematical content, but not as much as we would like to think. Other factors, such as arguments from authority and social intertia will play an important role.

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Both the Russell paradox and the Banach-Tarski "paradox" show that certain ideas are contradictory. It seems to me that the key difference between the two is that, in Russell's case, the ideas in question had been proposed (by Frege) as axioms for a foundation of mathematics, and they seemed sufficiently basic to be accepted, until the paradox appeared. In the Banach-Tarski case, one of the ideas involved in the contradiction is the idea that one can meaningfully talk about the "volume" of arbitrary sets in $\mathbb R^3$. (Here "meaningfully" is intended to include additivity and invariance under Euclidean motions.) Although that is a very appealing idea intuitively, I'm not aware of anyone's proposing it as an axiom (or even as a conjecture). The development of Lebesgue's measure theory had already shown that the intuition is not reliable and the measurability of general sets is a delicate issue.

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Although I am not so good with philosophical subtleties, I have always found useful to make a distinction between an antinomy and a paradox. The first leads to a formal contradiction, i.e., a logical inconsistency in your theory (you can prove both a formula and its negation).

The second `merely' defies human intuition, without being a (known) antinomy. Much less worrying (ask Frege :)).

Many just use `paradox' for both things, but I find this highly confusing.

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+ 1 –  Qfwfq Sep 27 '12 at 8:09

I don't see how these two a so fundamentally different. Russell's paradox tells us that we have to think more carefully about what a set actually is, and Banach-Tarski tells us that we have to think more carefully about what the measure of a set is and which sets are measurable.

When arriving at a counterintuitive statement, there are two possible conclusions: First, the previous intuition was wrong, and in this case the result is genuinely counterintuitive, and the second possibility is that the definitions or the logic were wrong and need to be changed. Banach-Tarski falls into the second category, because one would not conclude that matter can be created by clever cuts and rearrangements, but rather that one needs a thorough definition of measure.

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But the Russell paradox is a proof that a mathematical theory was inconsistent; whereas the Banach-Tarski paradox is just perplexing for us as human beings. –  Asaf Karagila Sep 26 '12 at 11:36
@Asaf: If we insisted that all subsets of $\mathbb{R}^3$ all had volume, Banach-Tarski would also become a proof that something is inconsistent. The point of "paradoxes" is not that they are formal inconsistencies (which depends entirely on what you accept as the axioms), but rather that they point to limitations in the logical structure of theories, be it theories of sets, measure or something else. In principle we are at liberty to propose whatever axioms we like, so it is not decided in advance whether a given "paradox" will be considered a proof of inconsistency or just an oddity. –  Andrej Bauer Sep 26 '12 at 14:28
And so, this is what the question is about: when faced with two apparently paradoxical statements, why do we formulate our mathematical theories so that one is indeed an inconsistency and the other just a really weird theorem? This is not a question within mathematics, but a question about mathematics. –  Andrej Bauer Sep 26 '12 at 14:30
@Andrej: I agree with that, but the Banach-Tarski paradox came up about 20 years after Vitali proved that it is inconsistent with choice to have all sets measurable. As I said that paradox in the case of Banach-Tarski is that it is inconceivable to us that one ball can be split into five pieces (one of which is a point if I recall correctly) and then reassembled as two balls of the same volume. –  Asaf Karagila Sep 26 '12 at 17:49

What you're describing as a "true paradox" is sometimes called an "antimony" and it means an actual logical inconsistency in the underlying theory. The Burali-Forti paradox is another example and it means there can't be a set of all ordinals (the ordinals are a proper class). By contrast, the Banach-Tarski theorem is consistent; it's just counterintuitive. The reason we don't hear much about "true paradoxes" (antimonies) these days is that logicians in the 1920's got the earlier inconsistencies under control, and (in all likelihood) we're not dealing with any actual inconsistent systems today, at least in everyday mathematics.

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Heh -- I think you mean antinomy. "Antimony" is a chemical element. –  Todd Trimble Sep 26 '12 at 17:55
antinomy :-) ${}$ –  Mariano Suárez-Alvarez Sep 26 '12 at 17:56
Antimony/antinomy came up here two years ago - see the comments on the Dick Palais answer to mathoverflow.net/questions/40920/… –  Gerry Myerson Sep 27 '12 at 0:25

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