Cardinality, Area, and ProbabilityDate: 09/05/2003 at 19:04:36 From: Giuseppe Urbani Subject: Cantor, cardinality and randomness the infinite Divide a rectangle into two parts, one with twice the area of the other. Pick a point randomly within the rectangle. It would appear that the probability of choosing a point in the larger part is twice the probability of choosing a point in the smaller part. But Cantor showed that every part of an n-dimensional space has the same cardinality. Therefore, shouldn't the probabilities be equal? Date: 09/05/2003 at 21:16:00 From: Doctor Tom Subject: Re: Cantor, cardinality and randomness the infinite Ciao Guiseppe, Your question is VERY advanced. To answer it in detail, you need to study something called "measure theory" which is usually taught in mathematics graduate school, or at the least, in your senior year at university. In the simplest case (say measure on a line), we would like to be able to assign measures to subsets of the real numbers satisfying the following conditions: 1) The measure of the interval [a, b] is b-a. 2) If S and T are two subsets of the line with no intersection, then the measure of (S union T) is the sum of the measures of S and T. In fact, we'd like more: If you have any countable non-intersecting subsets of the line then the measure of the union is the sum of the measures. 3) If S has a certain measure and you make a new set T that is the same as S translated along the line, then T will have the same measure. In other words, if you slide a subset rigidly along the line, it will continue to have the same measure. What would be great is if you could find a way to assign a measure to every subset of the real numbers that satisfies these conditions, but the first major result of measure theory is that it is absolutely impossible to do so! There are so-called unmeasurable sets for which there is no sane way to assign a measure. But this is only if you accept the axiom of choice to be true, so at once we see that the problem arises from a somewhat "obvious" axiom of set theory which turns out to be not so obvious at all! The same sorts of problems occur in any finite number of dimensions when we try to assign measures to n- dimensional Euclidean spaces for exactly the same reasons. There are some results that can be proven. For example, any countable set MUST have measure zero. But there are LOTS and LOTS and LOTS of sets for which it is impossible to assign a measure. The easiest example of an unmeasurable set, unfortunately, is not all that easy to describe. If you look at the rational numbers as a set, and then consider every possible set that is a translation of the rationals by a fixed amount, then you've divided up the reals into an uncountable number of disjoint subsets. Take one member of each of those subsets (this requires the axiom of choice) and form a new set that consists of your selections. That set will be unmeasurable. (Don't worry if this is hard to understand -- it certainly was for me at first.) There is a fascinating construction (that requires the axiom of choice) called the "Banach-Tarski paradox." Without telling you how to do it, here's what it does: The unit sphere in three dimensions can be divided into 5 disjoint subsets A, B, C, D and E such that if those subsets are rotated and translated properly and recombined, they form two complete spheres with identical size and shape as the original, with NO POINTS OMITTED in either! Clearly, at least some of the sets above must be unmeasurable, and it turns out that 4 of the 5 are. The set A is just the center point of the sphere. Anyway, I could go on for an entire semester in a graduate- level course on this, but at least I hope this is enough to get you interested and to do some reseach on your own. Look up "Banach-Tarski", the axiom of choice, and "measure theory" to get started. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ Date: 09/06/2003 at 07:26:14 From: Giuseppe Urbani Subject: Cantor, cardinality and randomness the infinite Thank you for your reply. It was very interesting, and I have started to do research on my own. However, I can't find in your reply the answer to my original question: How do we reconcile the difference in probability according to area with the equality in probability according to cardinality? Date: 09/06/2003 at 10:10:49 From: Doctor Tom Subject: Re: Cantor, cardinality and randomness the infinite Ciao Guiseppe, I guess the point of all that I said in the previous response is that the cardinality of the set does not determine its measure. There are some implications but they are not of much use: 1) In the real numbers, or in any higher-dimensional version of the reals, like the 2-dimensional plane or 3-D space (and in higher dimensions as well), every subset that is countable has measure zero. This is easy to show: Enumerate the points in the countable subset as p0, p1, p2, ... I can cover this subset with a larger set as follows: Given any real number E, choose the interval of size E/2 centered on p0. Then choose the interval of length E/4 centered over p1. Then the one of length E/8 centered over p2. Then of length E/16 centered at p3, et cetera. The measure of the union of all those sets is less than or equal to: E/2 + E/4 + E/8 + E/16 + ... = E So no matter how small a number E you choose, you can show that the measure of your countable set is less than or equal to that. Since E is arbitrary, the measure of the countable set is less than any positive number and the only such number is zero, so the measure of ANY countable set is zero. 2) Every set of positive measure is not countable (this comes directly from 1), above. In fact, even if the continuum hypothesis (another interesting axiom from set theory that states that the first uncountable cardinal is the cardinality of the continuum) is not true, it is still true that every set of positive measure has the cardinality of the continuum (I won't prove that for you). So EVERY set with positive measure has EXACTLY the same cardinality; namely, the cardinality of the continuum. And all the subsets of the reals that are NOT measurable also have exactly the same cardinality -- the cardinality of the continuum. So this basically means that if you know the cardinality of your set is less than the cardinality of the continuum, it has measure zero, but if your set has the cardinality of the continuum, it can have ANY measure, from zero to infinity. In other words, the fact that a set has cardinality equal to the continuum gives no information about its measure -- cardinality and measure are almost unrelated other than the fact that the cardinality has to be at least that of the continuum to get a positive measure. There are, by the way, sets with the cardinality of the continuum that have measure zero. The so-called "Cantor Dust" is such a set. It is obtained by beginning with the interval [0, 1] and removing middle thirds: first remove (1/3, 2/3) leaving the set that''s the union of [0, 1/3] and [2/3, 1]. Then remove the middle thirds of those -- the intervals (1/9, 2/9) and (7/9, 8/9), leaving four intervals. Then remove their middle intervals, et cetera. If you take the intersection of ALL those sets, it's clear that each one has measure 2/3 of the previous one, so the intersection has to be smaller than all of them, so its measure must be zero. But you can also show that there are an uncountable number of points left. This is done by looking at the base-three expansion of the numbers that remain, and showing that any number having a ternary expansion consisting of only 0s and 2s remains. But there's a 1-1 mapping of the ternary numbers with only 0s and 2s to the binary numbers having only 0s and 1s (just map the 2s to 1s) and those are ALL the binary numbers which clearly have the cardinality of the continuum. I hope this helps. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ Date: 09/06/2003 at 11:41:28 From: Giuseppe Urbani Subject: Thank you (Cantor, cardinality and randomness the infinite) Thanks for helping me see the distinction between cardinality and measure. I think I was instinctively confusing cardinality with 'amount', which was the cause of my confusion. |
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