Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. Wednesday, April 25, 2005 Main Idea: Expanding you understanding of ideas. Key Words: Countable, Uncountable, Axiom of Choice, Cardinal, Ordinal. Goal: Introduce some thoughts which while logical, are still puzzling. I. Counting Up to the number 5, people do not "count" the objects. They just know how many there are. You know a star on the American flag has five points. But it is difficult to know how many points there are on the star of the Australian flag. Beyond 5, one counts 1, 2, 3, .... until the set is exhausted, and the last number used is the number of objects in the set. When counting, (a) it is necessary to count every element. and (b) it is necessary not to count any item more than once, When counting coins, one can put them in a pile, and then count them as they are moved individually into a second pile. This requires a lot of control, a lot of space, and a lot of disturbance. For a summer job during the Cuban missile crisis, I worked in the Iowa Ordinance Plant making explosives. We were frightened over and over about how if we did one little thing wrong, the whole plant would blow up. Each room has a "maximum occupancy count". Our instructions were to never surpass that count. Even if the visitor was your boss, or your supervisor. If the visitor made the occupancy higher than the allowable limit, we were instructed to "point out the count, and if the visitor did not leave, we were supposed to leave." That made sense, until I found that my work station has a limit of 35 people, and there was no way that I could count them, much less keep a running total so that I would know whether I should leave or not. Technically, counting requires assigning an object to 1, an object to 2, and object to 3, etc. To do this one has to be able to identity the object enough to distinguish it from all other objects so that whenever you meet the object, you will know not only that it had already been assigned a number, but exactly what number has been assigned it. Now you can start to see the problem. If I have a gold fish bowl, I can see it has exactly two fish, but I cannot "count" them unless I can distinguish between the two fish. If they are identical, then I know there are two of them, but they are uncountable. You can take this process a step further by having the water murky. Occasionally you can see a fish as it swims near the side of the tank, but while you know that the number of fish is finite, you cannot count them. Of course, you could dump out the water and write numbers on the fish, but that corresponds to adding more axioms to what you are allowed to do. Replace the fish bowl with a forest and try counting deer, or crows, or rabbits. You probably can come up with an estimate on the size of the population, but this type of counting is not the concept of "countable". Two sets "have the same number of elements" if there there is a 1-1 map between them. The word for "number of elements" is "Cardinality". It is indicated by absolute value bars. | A | = the "Cardinality of the set A" = the number of elements in A. Thus | {A, B} | = | {1,2} | because one can map A <---> 1 B <---> 2 for one example. This becomes interesting in infinite cases. |{1,2,3,...,n,...}| = | {2,4,6,... 2n, } | A set and its subset can have the same cardinality. What is the 1-1 mapping. The cardinality of the positive integers is called w (little omega). A set has w elements when it can be put in 1-1 correspondence with the positive integers. (1) {1,2, ..., w} is countable. How? (2) The points (i,j) in the first quadrant is countable. How? (3) The lattice points {i,j} of the whole Cartesian plane are countable? (4) The rational numbers are countable. How? The cardinality of the real numbers is called c. The number c is also the cardinality of any open interval. How? It is also the cardinality for any closed interval as well, but the proof is a trifle more complicated. The number of points on the plane is also c. How? There is an easy "almost airtight" proof of this, but it takes a bit of tweaking to make it completely correct. ............................................................... Now we go to "ordinal" numbers Ordinals refer to order. They are defined by comparison to the integers. Let o be the empty set. The integers are 0 1 2 3 n+1 w w+1 o {o} {o,1} {o,1,2} ... {o,1,2,...,n} {o,1,2,...n,...} {o,..,w} this list just keeps growing. The next number is the set of all numbers smaller than it. 2 w 1 ... w, w+1 w+2, ... 2w, 2w+1, ... 3w, 3w+1, ... 4w ... w ... w ... _()_ <---------------------------------These are all countable-------------------> First uncountable ordinal Big omega If you start any where on this "long line" and walk backwards, you will reach the beginning in a finite number of steps. If you start to the left of Big Omega and walk to the right, you will never approach Big Omega as a limit. You will always have a limit short of Big Omega. The axiom of the continuum is that "the cardinality of the real numbers is the same as the cardinality of the first uncountable ordinal". It has been proved that this one cannot prove it either way. If you assume c is the first uncountable number, there will be no contradictions. The real numbers are uncountable. I am giving the standard proof which I do not think is airtight. But just because they are uncountable, it does not mean that there are MORE of them than w. It is more subtle than that. There are "countable" sets which have subsets that are uncountable. The set of all touring machines is countable, but the subset of them which halt is uncountable. There is the "inside-outside" phenomenon. They are uncountable from the inside, but countable from the outside. With any system in mathematics, there are certain allowable things you can do. For example, if you have sets A and B, one presumes that you can create the sets A U B and A n B. Using the information available, you want to know what can be created. Inevitably, you will run into problems because you will not have enough words. The typical result is that one extends the number of allowable constructions. And then extends them again, and again. Every model of set theory contains the integers, and also a copy of the rationals and the reals because we can build the reals up out of the positive integers. Inside any model of set theory, the real numbers will be uncountable. But, you can have a model of set theory where everything in the model, real numbers and everything else, is countable, from the outside. I see this as a problem with not having enough words for distinguishing objects. If you cannot distinguish things, you cannot count them. You can approximate the count, which is the same as saying, "many many" which is interpreted to be more than you can distinguish. Which brings up the Axiom of Choice. The Axiom of Choice says that given a collection of non empty sets, that you can pick out one element from each set. Pick out means that you can't just reach in a grab an element. If the set is presented a second time, you have to select the same element. And you cannot somehow mark the element the first time. Using the axiom of choice, you can count the elements in any set. I believe that assuming the axiom of choice means that you must have a very small system and most likely the whole system will be countable from the outside. Of course, if it is countable from the outside, the Axiom of choice is possible by simply picking the first element that occurs in any set. In calculus or differential equations, there has never been (as far as I know) anything useful in which the Axiom of Choice, or the Axiom of Continuity made any difference what-so-ever. If a series converges assuming the Axiom of Choice, it converges without assuming the Axiom of Choice. There are collections of non-empty sets from which no one knows how to pick an element. But whether you assume that you can, or whether you do not assume any such thing, you will not get anything of interest. My personal belief is that the real numbers are countable. I think the diagonalization process is self referencing which makes it a no-no. Now this is what I prefer to think of as the real numbers. I tend toward the philosophy like Dedekind, or Cauchy. The real numbers are those that can be captured in some way. I believe these types of reals are countable. Others say that there are many many more, hidden about inside the real number line, but one can never find them, or even know that they are really there. In particular, there are holes in the real number line, but you cannot locate them, nor can you find a Cauchy sequence that approaches it. Thank you for taking the ICN ISU calculus sequence. I am glad that ISU offers this option to students who are ready for calculus, but not part of a big enough group to justify a class locally. I really appreciate the assistence that I get from your teachers. Without them, this program just would not run and I believe we would have lost a valuable part of our educational system. Please thank your teacher today for their part in making this opportunity happen. Class is over. I will send out your grades today.