# Your most counterintuitive yet simple math problem

Mine is the Monty Hall paradox. For an introduction, please see http://en.wikipedia.org/wiki/Monty_Hall_problem" [Broken]

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How about the probability that two people share a birthday in a group of 23 people?

It is surprisingly just over 50%. 23 is the smallest value for which the probability is over 50%.

Perhaps even more surprisingly, if there are only 50 people in a room, the probability is 96.5% that two people share a birthday.

Most people would guess a much lower value.

That one is great too, George. It always amazes me. (Great to try out on a third grade class!) I suppose that it could be phrased in any number of contexts.

HallsofIvy
Homework Helper
How about the Banach-Tarski property: that a sphere in three dimensions can be partitioned into a finite number of subset and, by rigid motions reasssembled into two spheres of the same size as the first sphere.

How about the Banach-Tarski property: that a sphere in three dimensions can be partitioned into a finite number of subset and, by rigid motions reasssembled into two spheres of the same size as the first sphere.

Do you understand this one? I don't... Not one bit. A friend told me about it and I immediately told him he had been misinformed. But he send me a link later and I was baffled... It is obviously illogical and it is a genuine paradox!

What is the explanation... It can't be done in real life, so what is the error?

edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

No, the incompleteness of mathematics has nothing really to do with it. It has everything to do with infinite sets and the way you assign a "volume" to them.

The basic idea that the Banach-Tarski "paradox" is based on is that, say in 3D Euclidean space, you can assign a volume (a measure) to certain subsets of the space. However, any reasonable definition of a measure must have some sets that are unmeasurable; they really don't have a well-defined volume. The idea is that the ball can be split up into a few subsets that are unmeasurable; you can then transform these unmeasurable sets (by rotations and translations) such that they don't overlap, and then take their union to obtain another measurable set, which is in fact two balls of the same size as the original! The result depends crucially on the fact that you're in 3D space; it doesn't happen in 2D space. It also depends on the fact that unmeasurable sets are involved; the measure of a (countable) union of disjoint measurable sets is the sum of the measures of the original sets.

This isn't possible in real life, of course, as in reality balls are composed of finitely many atoms.

You may or may not gain some more insight by looking at Wikipedia.

Fredrik
Staff Emeritus
Gold Member
edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?
It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set $X=\{Y|Y\notin Y\}$ (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

The Banach-Tarski paradox is just a result of the fact that it isn't possible to define a "size" of an arbitrary subset of the real numbers in a meaningful way.

Monty Hall paradox can be studied with cut up pieces of individual paper, but a very convincing way I use is to consider the three cases for door A, which is arbitrary and covers all cases:

We have: Behind A, Behind B, Behind C. Only in 1/3 of the cases does it pay not to switch. In the other two cases, since one of the wrong doors has been eliminated, by switching we have the right door!

It helps to draw a very simple grid indicative of that, and the answer is obvious. But, attempting to work it over in your mind, doesn't work so well as a simple diagram.

It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set $X=\{Y|Y\notin Y\}$ (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

When I read his statement I was thinking more of Gödel's incompleteness theorems.

For me it was understanding that there are more reals in (0,1) than all rationals

The fact that there is a real to real function that is continuous, not constant on any interval, and has uncountably many zeroes.

It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set $X=\{Y|Y\notin Y\}$ (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

Yeah that's what I was thinking of. So until that paradox was discovered, where there no restrictions on what sets were allowed to form?

Hurkyl
Staff Emeritus
Gold Member
Yeah that's what I was thinking of. So until that paradox was discovered, where there no restrictions on what sets were allowed to form?
Yesish. Keep in mind that set theory had only recently been made explicit, and that there are obviously other restrictions (e.g. sets have to be sets). But in its initial form, set comprehension was indeed unrestricted.

HallsofIvy
Homework Helper
Do you understand this one? I don't... Not one bit. A friend told me about it and I immediately told him he had been misinformed. But he send me a link later and I was baffled... It is obviously illogical and it is a genuine paradox!

What is the explanation...
As Adriank said, it involves using non-measurable sets

It can't be done in real life, so what is the error?
There is no error. Perhaps your "real life" is too restricted!

edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?
The fact that "naive set theory" is not rigorous only means that we need a more sophisticated concept of "sets"- and that has already been developed.

If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!

Borek
Mentor
If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!

As I have stated on numerous occasions I am mathematically challenged, so I can be wrong, but from what I understand it says that even if we will do everything, there will be still statements that we will be not able to say if they are true or false.

Could be that implies that there is still something to do outside of the system in which we can't decide...

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Ooo, let's not forget that most real numbers are uncomputable.

I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?

Here's something for your noggin. I don't know your familiarity with set theory, but I hope you can enjoy it.

By http://en.wikipedia.org/wiki/Cantor%27s_theorem" [Broken], the number of subsets you can make from a given set will always be strictly greater than the number of elements in the set itself. That is, if S is a set and P(S) is the powerset of S, S < P(S).

Take A to be the set of all sets, defined by A = {S | S is s set}. By Cantor's Theorem, A < P(A). However, P(A), the powerset of A, is also the set of all sets. So it's cardinality is both equal to itself and inequal to itself. A paradox.

The resolution to this paradox is something similar to what your comment above hints at. It's not a fundamental flaw, inasmuch as it's not really a flaw. Most mathematicians don't need to care about it at all, and boring old set theory will almost never get you into trouble as long as you're not looking for it.

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D H
Staff Emeritus
That an object with finite area can have an infinite perimeter.

For me it was understanding that there are more reals in (0,1) than all rationals
That's a good one, too. $2^{\aleph_0}>\aleph_0$ is a mathematical equivalent to relativity theory in the sense that both are crackpot magnets. Now that Fermat's last theorem has been put to bed, the majority of "interesting" unsolicited proofs that professors of mathematics receive from the lay community are attempts to disprove Cantor's result.

That an object with finite area can have an infinite perimeter.

Are you talking about fractals/space filling curves? What you said made me think of something else, objects with infinite area but finite volume (Gabriel's horn is the one I was thinking of).

D H
Staff Emeritus
Sorry for not being clear. An unbounded planar object can clearly have finite area. Since it is unbounded, it obviously has an infinite perimeter. So yes, I was talking about fractals -- e.g., the coastline of Britain.

Fredrik
Staff Emeritus
Gold Member
I think I read somewhere that there's a continuous bijection from $\mathbb R$ into $\mathbb R^2$. (I hope someone will let me know if this is wrong). The existence of a bijection isn't surprising to me. Intuitively, that just means that the two sets have "the same number" of members. The existence of a continuous bijection is very counterintuitive however. That would be a curve that goes through every point in a plane without ever intersecting itself.

Yes, $$|\mathbb{R}| = |\mathbb{R}^{n}|$$ I believe

Well, there is a continuous surjection from [0, 1] to [0, 1]2, but no continuous bijection. The existence of such a bijection would mean that [0, 1] and [0, 1]2 are homeomorphic, which isn't true.

This would be my pick for the counterintuitive math thing. :)

For me it was understanding that there are more reals in (0,1) than all rationals

There are more reals between any two different rationals, than all rationals overall.

This is perhaps a slightly more interesting concept. Although it is essentially the same.

Besides, this is an unusual definition of "more" anyway. It is not the same sort of "more" that we usually use, all that we really know is that there isn't a way of counting (or pairing with the naturals) all the irrationals.

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Yes, you have.