What's the weirdest theorem you can think of?

For years mine was Euler's , and it sort of still baffles me, but maths being fab as it is, I've seen it from enough perspectives that it sort of makes sense now: just calculate numerically (knowing that the taylor series converge) and observe the rotational properties of multiplication by the result. The rest is just the continuous interpolation of exponentiation at work.

Even Godel's masterpiece is just a statement that the natural numbers are complicated enough to encode talk of any system that can produce them.

Are there any truly counterintuitive theorems, or are we just looking at them the wrong way? What's the best can pass for pathological in this age of axiomatic sidestepping?

Answers on a to the usual address.

Surely the Banach-Tarski Theorem (I maintain `paradox'!)

More down-to-Earth, I don't like the way holomorphic functions are specified by their values at the boundary, by the Cauchu Integral Formula. I feel that surely I ought to be able to continuously deform it a little bit and have it still holomorphic!

In a sense it's a hard question to answer because you generally "get used" to a lot of counterintuitive facts in maths and therefore don't find them as strange anymore, especially if you understand the proof well enough. Besides that, your intuition of things changes the more you study maths, so things like the fact that and have the same cardinality could be labled as counterintuitive (because when I first saw them, my intuition was that it shouldn't be the case) even though I would consider similar statements perfectly "intuitive" now.

I do find the Banach-Tarski paradox counterintuitive from that point of view, but it doesn't bother me in any particular way (in fact I find it pretty cool that it works). In fact, using the axiom of choice (and in particular the well-ordering principle, which I also find pretty damn counterintuitive), it's not too hard to construct a lot of strange/unintuitive mathematical objects, e.g. a set that intersects every line in the plane in exactly two points (just well-order the lines of the plane and pick two points from the first, then two points in the second not in the first, etc. by transfinite induction; at every step you will have chosen less than points and therefore you can go on - it's also an interesting side note that it is unknown whether this can be done without AC or even if you can find a Borel set that satisfies that condition).

Other theorems I find more or less counterintuitive are, for example (and probably missing a few good ones), Tarski's method of "squaring the circle" (which is quite similar to Banach-Tarski), the existence of a polynomial in 26 variables whose positive values are exactly the primes, Goodstein's theorem or even the fact that almost all real numbers aren't computable, but, again, most of these make perfect sense once you understand the principles behind them.

I'm not sure whether I would label Godel's incompleteness theorems as counterintuitive because I think the surprise most people experience when they first see them doesn't, strictly speaking, come from mathematical intuition but a sort of wishful thinking about how axiomatic systems (that are often in some ways too complicated for our intuition) should behave. What I find somewhat amusing though is the fact that a lot of people (and in particular non-mathematicians) tend to take Godel's theorems to mean that no axiomatic system is complete/"provably consistent" and hence are surprised by the fact that some of them are (which in a way is like they would find that fact counterintuitive).

Ok,

Not that much of a counterintuitive idea, but... I do find that it strange/amazing that the Stargate Theorem can be solved by induction... I thought it was supposed to be really difficult to prove (I can't even remember the statement of the problem), but then was told that it can be done by doing induction on theta. I remember staying up working past 10pm one night (and you shouldn't work after 10...) trying to figure out just how the induction worked... I couldn't even really understand the base case when theta is 0!

Regards.

Yeah, it can't remember the exact statement but I think it was something along the lines of: the sequence , where contains over 9000 primes. This is supposed to have huge consequences for twin primes, the Erdős-Tsáot-Woodin conjecture and possibly even Goldbach. Induction does seem like a sensible approach, even though I personally haven't tried it yet.

One other puzzle about the SG theorem which bothers me is why we use theta as opposed to a more normal letter for the induction... Isn't it more traditional to use n/m/k types of letters [I am aware that strictly this clearly has no effect on the actual result, i just find it strange]... Perhaps it comes from the fact that theta looks slightly like the stargate from that film... But from what Cosmin and my tutor told me this theorem preceeds the film, and has been called Stargate since conception.

Hmm.