Banach-Tarski Paradox:
--I studied set theory in my undergraduate days, liked it, and I liked this paradox. But just recently began playing around with any implications it may have on libertarian legal theory--
For non-mathematicians let me quote Wikipedia:
The Banach-Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, 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. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun".
As well as from http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml
Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox.
So why can't you do this in real life, say, with a block of gold?
If matter were infinitely divisible (which it is not) then it might be possible. But the pieces involved are so "jagged" and exotic that they do not have a well-defined notion of volume, or measure, associated to them. In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"! (Or else the above example would show that 2 = 1.)
An alternate version of this theorem says (and you'd better sit down for this one): it is possible to take a solid ball the size of a pea, and by cutting it into a FINITE number of pieces, reassemble it to form A SOLID BALL THE SIZE OF THE SUN.
Presentation Suggestions:Students will find this Fun Fact hard to believe. You might want to say that mathematics in this case reveals to us that we must be very careful about how we define things (like volumes) that seem very intuitive to us.
The Math Behind the Fact:First of all, if we didn't restrict ourselves to rigid motions, it would be more believable. For instance, you can take the interval [0,1], stretch it to twice its length and cut it into 2 pieces each the same as the original interval. Secondly, if we didn't restrict ourselves to a finite number of pieces, it would be more believable, too: the cardinality of the number of points in one ball is the same as that of two balls!
The proof involves studing group actions on the sphere, specifically, subgroups of the rotation group "SO(3)" that are free subgroups on 2 generators. Such strange subgroups allow one to construct "paradoxical" sets: sets which are congruent (under the group actions) to 2 or more "copies" of themselves! The proof also depends on the Axiom of Choice.
Some Ramblings
The reason why such a system is both incomprehensible and beyond reality is because there is nothing in our universe of infinite density (outside of theory). For example, black holes have zero volume and infinite density, but such a singularity is still only theoretical. Yet, if such an item were in existence it would have the theoretical capability (but maybe not the actual capability) of being distributed in any quantity you like. Would there be any value for an object such as this? I know we prefer more to less, but only when we value such an object over another object right?
Finally, given all of this blather, what about the market of information? Information is without volume, but is it infinitely divisible? Can we say that information is something of infinite density and therefore able to be divided and copied infinitely?
I attribute whatever this rambling is to my six pack of Samuel Adams Winter Ale.
Yet, what about information? Information exists
You can't partition a word or impress it onto your memory multiple times because it may only exist in one place.
Read until you have something to write...Write until you have nothing to write...when you have nothing to write, read...read until you have something to write...Jeremiah
The incomprehensibility shows nothing of the sort. What is shows is that we mathematicians can take a good idea (instead of working with the specifics of this case, I'll abstract the pattern) and ruin it by taking it too far. Set theory has the implicit premise that V is the universe, although most of us working in the field or in areas touching the field soften it to V is the set-theoretic universe, making it reasonable to wonder why one cares about undecidable things. Back in the real universe, ZF+CH seems reasonable as a set of true statements, ZFC does not. The whole point of choice is that it allows one to create functions which one cannot construct by any imaginable process. Choice is not needed for finite or countable collections, which also happen to be instances where it seems perfectly reasonable. (A weaker form of choice is used for countable collections, but this weaker form does not imply BTP.)
Set theory began with the investigation of reasonable things, which had real applications. It is reasonable to wonder about the size relations between sets of numbers, and this has real applications in analysis, and hence probability. It is reasonable to extend some essential properties of the natural numbers upwards and to wonder if this construction gets you all the sets you use in math, hence infinitary combinatorics and cardinality, and even the discussion around MA. After that, you get into taxpayer dependent mathematics, that is, math that would not have developed if not for the fact that mathematicians do not answer to real employers, but rather are supported either by taxpayer money or corporate money - and corporations receive handouts from government. What's more, they are safeguarded by tenure, from ever having to justify the value of their research. Thanks to our institutionalized educational system, specialists send papers back and forth to other specialists, who make peer-review decisions. Hence, the peer-reviewer essentially says "well, yes, this game is quite interesting to me, hence I'll say 'this is a deep result with important ramificiations in the theory.' " On the basis of such decisions, tenure is granted. Not everything that develops in math, thus, has meaningful implications. A bunch of taxpayer funded mathematicians in Canada decided to study zombie apocalypse to see if the movies have it right. For this the government takes your money and says "it's for education, don't you like education!" Meanwhile, the academics consider it a hassle to have to actually, you know, teach students.