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?

• @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/… Commented Sep 26, 2012 at 8:44
• 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. Commented Sep 26, 2012 at 9:43
• Voting to close as "not a real question" (not even from a metamathematical or philosophical viewpoint) Commented Sep 27, 2012 at 8:05
• @TobyBartels, is it acceptable to be amused that your 2-part comment on "Banach–Tarski" accidentally used the term twice consecutively? :-) Commented Dec 11, 2018 at 16:46
• @LSpice : Yes, it is; although after 6 years, I can no longer remember whether or not that was really an accident. Commented Dec 11, 2018 at 21:53

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.

• Since this is bumped to the top: I’m sure you’re aware of this Andrej, but for any other readers there are folks exploring changing the underlying logic to ‘get used to’ inconsistency in a certain sense. Commented Nov 20, 2023 at 13:44
• I believe that would be the "We might change rules of logic, definitions, or axioms, everything is up in the air" part of my answer. Or are you talking about something else? Commented Nov 20, 2023 at 14:19
• Yes, this is one example of what I thought you meant with that reference; I thought a link to explore might be of interest to future readers. Commented Nov 20, 2023 at 14:23
• @AlecRhea: of course. Commented Nov 20, 2023 at 14:28

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.

• Well, these: en.wikipedia.org/wiki/Solovay_model are models of ZF chosen specifically so that all subsets of $\mathbb R$ have a defined volume. Commented Jun 2, 2016 at 18:56
• @jcast Yes, and the nature of those models reinforces what I wrote. One can meaningfully talk about the volumes of the subsets of $\mathbb R^3$ in such a model, but those are not arbitrary subsets of $\mathbb R^3$. In fact, the sets in the model are all definable (in the language of set theory, using countable sequences of ordinal numbers as parameters) in a larger model of ZFC, where there are non-measurable sets. Commented Jun 2, 2016 at 23:29

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.

• + 1 Commented Sep 27, 2012 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.

• 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. Commented Sep 26, 2012 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. Commented Sep 26, 2012 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. Commented Sep 26, 2012 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. Commented Sep 26, 2012 at 17:49
• I like what I hope I may call @AsafKaragila's implicit rephrasing: if the Banach–Tarski paradox is a paradox, why isn't the Vitali construction a paradox? Commented May 10, 2023 at 13:58

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.

• Heh -- I think you mean antinomy. "Antimony" is a chemical element. Commented Sep 26, 2012 at 17:55
• antinomy :-) ${}$ Commented Sep 26, 2012 at 17:56
• Antimony/antinomy came up here two years ago - see the comments on the Dick Palais answer to mathoverflow.net/questions/40920/… Commented Sep 27, 2012 at 0:25
• @‍DickPalais's answer referenced by @GerryMyerson. Commented May 10, 2023 at 13:52