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40 Answers 40

Perhaps the Banach–Tarski paradox: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e. subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape.


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Already quoted. –  Giorgio Mossa May 2 '12 at 20:08
... in ineff's answer. –  Martin Sleziak May 6 '12 at 7:30

How about the Löwenheim–Skolem theorem?

One of its consequences is that the field of real numbers has a countable model.

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Not exactly. The completeness property is second order. The countable model will be a real-closed field, but not complete. –  Asaf Karagila May 8 '12 at 4:38

The Weierstrass function. It showed that a function can be continuous everywhere but differentiable nowhere. This was (and still is) counterintuitive.

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I am surprised nobody has given this example: It was once thought that continuous functions have derivatives at almost all points (as other answerers have used). Then comes the Brownian motion model that generates a random curve, which, with probability 1, is a curve without derivative at any points!

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I always found the main correspondence of class field theory counter-intuitive. Somehow the (Abelian) field extensions of a number field $K$ correspond to arithmetic objects in $K$ itself.

You wouldn't think that the possible ways of extending a field (albeit in a nice way) would just depend on the field you started with.

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For quite a while, when I was very young, I thought that the solution to the famed brachistochrone problem (which asks about the shape of a wire where a bead sliding frictionlessly on it under the influence of gravity will take the least amount of time to finish) was not a straight line was pretty counterintuitive, after being repeatedly told that the straight line is the shortest path between two points. (To spoil: the true brachistochrone is an inverted cycloid, which is the curve traced by a fixed point on the rim of a rolling circle.) I went through the derivation a number of times before finally being convinced.

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Although Dan Brumleve and plm linked the list of paradoxes, the famous Braess's paradox deserves a special attention.

"For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times."

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I find it counterintuitive that the rational numbers have zero measure. This means that for any $\epsilon$, no matter how small, we can find a collection of intervals that includes every rational but whose total length is less than $\epsilon$.

Such a covering includes all the rationals, but somehow must omit nearly all the irrationals. The question has some up here several times about what irrationals are missed (1 2 3), and Asaf Kargila recently described the result as “baffling”. So I am not the only person who is surprised by this.

Von Neumann famously said that in mathematics you don't understand things, you just get used to them. This for me is one of those things.

(The countability of the rationals, or the uncountability of the irrationals, may be similarly baffling, but I have gotten so used to both that I no longer find either one baffling, and I am not sure which one should be considered counterintuitive.)

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It's fairly straightforward to prove that the Cantor set is uncountably infinite (i.e. it has the same cardinality as the real numbers). You would think that such a set would have at least positive Lebesgue measure. WRONG-IT ACTUALLY HAS MEASURE ZERO! This blew my mind when I first learned about in undergraduate real analysis and it's why it puzzles me why Paul Halmos famously described measure theory as "a generalized kind of counting". This example seems to show that measure has nothing to do with cardinality, which is what most of us think of as counting in abstract sets!

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The standard Lebesgue measure definitely has something to do with cardinality. If a set has positive measure (or is unmeasurable) it has the same cardinality as the reals. Measure is a more refined sense of counting-so refined, in fact, that it is only really interesting on a class of sets all of which have the same cardinality. –  Logan Maingi May 4 '12 at 7:56
The existence of "counting measure" on any set seems to give one indication that measure theory is a generalized kind of counting. –  Pete L. Clark May 5 '12 at 17:52
@Pete: I think he was trying to ping you. –  The Chaz 2.0 May 8 '12 at 4:00
If instead of taking away the middle 1/3 at each iteration, you instead take away the middle 1/4th, you end up with a "fat" Cantor set which, surprisingly, has positive measure. –  Nick Alger May 16 '12 at 2:07
@LoganMaingi: Solovay showed that it is independent of ZFC whether there are nonmeasurable sets with cardinality less than $2^{\aleph_0}$. See this answer of JDH for a survey of related results. –  Jonas Meyer May 17 '12 at 3:40

Independence in Statistics. If I have a box with red & black socks, and I take one, and then another one the probability of both socks being red is sometimes different from the case when I, instead, pick a pair. (EX.If all socks are connected to every other with a string, like a complete graph, and then I pick some string, automatically choosing the socks at either end as the pair selected).

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