The standard introduction to transfinite theory is fraught with paradox and muddled dialectics. In this article, I suggest a different direction for approaching the interesting issue of sets, continuity, large sets and infinity. I've titled the approach "rich theory".
The main gist of rich theory is that the behavior of large sets is a fertile and worthwhile area of investigation. There are many different ways to approach the issue. Rich theory is in keeping with the goals of Descriptive Mathematics in that the primary goal of the theory is to introduce students to the language used to describe large sets.
For example, rather than trying to explain the nature of continuity, the teachers of rich theory simply try to introduce the concept of continuity, and the problems we have in trying to master this illusive topic. The primary goal of this approach is to build the student's vocabulary and basic understanding of the subject. The goal is not to establish the teacher as a guru, or to serve as a foundation of a theory of everything.
There is nothing startling or new about rich theory. I chose the name simply to indicate that mathematics is a rich subject, and that there are many different ways to approach complex subjects. Too much of mathematical and philosophical writings are dedicated to trying to find fault in the works of others. The truth of the matter is that there is not a simple single way of thinking.
The most fruitful approach to the infinite is to find the gems of wisdom from all the thinkers on the subject. It is my hope that this outline could be modified into a successful presentation of the large sets and the infinite than the diagonal method.
The study of the infinite reaches into antiquity. When speaking of infinity, Zeno of Elea often jumps to mind. Zeno's paradoxes defied resolution for several millennia. You will still find logicians today claiming that the paradoxes have never been adequately resolved. Undoubtedly, thinkers of tomorrow will find interesting new twists to the theory.
Perhaps the most significant ancient confrontation with the infinite came from within the ranks of the Pythagorean Society. Legend tells us that the Pythagorean Society was more than a simple group of intellectuals, and that they had elevated their belief in numbers to the status of a religion.
The Pythagoreans saw integers and ratios as the building blocks of the universe. Purportedly, the society unraveled with the discovery that it was impossible to express the of square root of two as a ratio between two numbers. The discovery of irrational numbers was both a major intellectual and social crisis.
While the ancients enjoyed dabbling in the mystical, most Greeks were pragmatic at heart and quickly tired of the paradoxes and circular reasoning involved with the infinite. After the demise of the Pythagoreans, Greek mathematicians turned to Geometry and Logic, which they felt they could explore without getting twisted in the circular logic of the infinite.
The Greeks and Romans still allowed reducio ad absurdum arguments; however, you can see a decided effort to avoid reference to the infinite in many of the ancient Greek and Roman writings.
Hence, I characterize the ancient view of infinity as a subject to be avoid. This view essentially holds that infinity is nothing. It is an incomprehensible that we should avoid referencing.
In the 16th century, Galileo contemplated the infinite, and decided to compare the size of the natural numbers and squares. He observed that, if you placed the set of natural numbers in a one to one correspondence with the squares, you would not run out of natural numbers before squares. He concluded that the sets were the same size. Since all infinite sets are endless, it is easy to extend Galileo's idea and conclude that, since all infinite sets are endless, all infinite sets are the same size.
From this perspective, infinity is a singularity. It is the one great all.
The most important event in the history of the infinity occurred in the 17th century when Leibniz and Newton independently discovered the fundamental theorem of the Calculus. To this day, Calculus stands as the crown jewel of western mathematics. Newton's Calculus as the basis for our understanding of physics, and the theory plays a primary role in almost all areas of higher mathematics.
The discovery of the fundamental theory of Calculus, and the great advances in physics and engineering that followed the discovery created a tremendous problem for mathematicians and logicians. The crown jewel of western mathematics did not have an adequate logical foundation. The formulas for Calculus worked, but the subject invoked references to infinitesimals and infinite numbers that that logicians had rejected as suspect.
It was not until Cauchy solidified that modern definition "limit" that mathematicians felt comfortable with the foundations of the theory.
The next great event in the history of the Infinite came from the pen of Georg Cantor. Cantor's primary interest in the infinite was the nature of continuity. The Rational Numbers are obviously discontinuous. You can find irrational numbers between any given two rational numbers.
Mathematicians have speculated that you can represent every single point on a line with an infinite decimal. They have further speculated that this set of infinite decimals is somehow continuous. The next question is simply: What fundamental properties differentiate the continuous real numbers for the discontinuous rational numbers.
Georg Cantor was a disciple of Immanuel Kant. Like other German Idealists in 19th and early 20th century Germany, Cantor was interested in casting aside the chains of Aristotelian logic and to develop a new dialectical method based on oppositional logic, etc..
Cantors assumption is that there is something about real numbers that makes the set continuous. The set of rational numbers is big, but discontinuous. Cantor worked to establish a dichotomy between the rational and reals. He used his famed diagonal method as a way to demonstrate this dichotomy. As such I characterize the Cantor's view of infinity as a duality. There are two types of infinities. There is a discontinuous and continuous brand of infinity.
And so I complete my short, woefully incomplete biased characterization of the history of the infinite.
Rich theory sees the infinite as an infinitely rich topic. Each of the above world views have great deal of merit. Infinity often appears as nothing. Divide one by infinity and you have something near zero. 1/∞ approaches 0.
All infinite sets are endless in nature. Infinity behaves as a singularity. When we consider the distinction between discontinuous and continuous infinite sets, we can see the dualistic nature of infinity.
All the above views have merit. Infinity is a rich subject. None is completely correct.
The challenge for mathematicians isn't simply to decide which view is right or wrong. It is to find the most productive way of presenting the subject to the student. I believe that the most effective approach to begin with arbitrarily large sets.
An arbitrarily large set is not infinite. It is simply a set that is so large that we can pretty much perform basic mathematical calculations without having to worry about the upper limits of the set.
Computer scientists make regular use arbitrarily large sets. In designing a computer program, programmers decide how much room to allocate for variables. A great example of this is the humble IP address used on the Internet. The designers of the Internet chose to use a 32 bit integer to identify the computers on their network. A 32 bit integer gives 4,294,967,296 possible addresses. Four billion seemed like a sufficiently large address space in the 60s and 70s. Programmers of yesteryear couldn't even imagine the success of the Internet. The 4.2 billion addresses reserved for Internet addresses is not sufficient in a world with six billion people. It fails to consider that people want IP addresses for their toasters.
The world has exhausted its pool of 4 billion addresses. Today, the W3C hopes to move the Internet to a new standard called IPv6, which uses a 48 bit address. The 48 bit address has 2.814 * 10^14 addresses.
The Internet provides an example where scientists chose a large upper limit for the size of a variable. The program worked for a long time. Today they need upgrade.
Y10k will provide similar challenges when all these computer programs designed for four digit years fail. I have no reason to think that the programs I am writing in 2003 will be running in 9999. I know for certain that they will crash and burn in 10,000.
Digital computers are discrete by nature. A digital computer cannot hold a full infinite series. The scientists who work with computers have, of necessity, become masters in the use of large sets. The study of large sets would have appeared as a fanciful topic for idle speculation in Cantor's youth. Today, however, using large sets is routine.
I am writing this article with 196 megabytes of memory and 5 gigabytes of storage, and it is considered wimpy by today's standards. I can buy a computer with 5 times the capacity for a third the price.
A good way to start the study of large sets is perform a few simple operations on a set. Let's say we have a set with n members. n is a really large number. Adding or subtracting one unit from the set doesn't have a big impact on the total size of the set. Addition has a minor impact on the size of the set.
Multiplication is a different story. Multiplying a large set by two doubles it. Removing one tenth of the units in a set decimates it. Dividing the set by two halves it. Multiplication has a much larger impact on the size of a set than addition.
Squaring a large set has a much bigger impact than multiplication. Exponential functions, such as 2^n, have an even larger impact.
In studying large sets, we see that different operations have different impacts on the size of the set. You end up with a very interesting layering of operations. We can see this a little bit clearer if we examine the power set.
This layering provides us with interesting insights into issues like dimensions.
| Cross Sections |
|---|
|
In the cross section proofs, I list the cross sections from left to right in the following manner: 1 3 6 a f ... Producing: 1,2,3,4,5,6,7,8,9,a,b,c,d,e,f... |
In performing the proof, you need to decide if you care about order, and whether or not you want to exclude repeated elements in the set. You may not want {1,1} in the enumeration. Likewise, you might consider {1,2} to be the same set as {2,1}. Since space is not a primary consideration in this exercise, I won't worry about the duplicates. I will be working with ordered pairs, ordered triplets and so on, instead of proper sets.
In the first step of the exercise, I list the set of ordered singles in a row. As expected the set of ordered singles isn't all that interesting. I mean, how much ordering can you do when you have only one thing? Here is my creation:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)...
To list the set of ordered pairs, I put the set of ordered singles in the header of a table, and add a new element in each of the rows as follows:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ... 1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1) (9,1) (10,1)... 2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (7,2) (8,2) (9,2) (10,2)... 3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (7,3) (8,3) (9,1) (10,1)... 4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (7,4) (8,4) (9,4) (10,4)... 5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (8,5) (9,5) (10,5)... ...
Taking cross sections of this table, I create an ordering of ordered pairs that looks like:
(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4) (2,3) (3,2) ...
You can create a listing of positive rational numbers from the listing of ordered pairs by calling the first element the numerator and the second the denominator. You would probably want to simplify the fractions and remove duplicates.
To create the list of ordered triplets, I put the ordered pairs in the headers of a table and a new element in each of the rows, as follows:
(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4) (2,3)... 1 (1,1,1) (1,2,1) (2,1,1) (1,3,1) (2,2,1) (3,1,1) (1,4,1) (2,3,1)... 2 (1,1,2) (1,2,2) (2,1,2) (1,3,2) (2,2,2) (3,1,2) (1,4,2) (2,3,2)... 3 (1,1,3) (1,2,3) (2,1,3) (1,3,3) (2,2,3) (3,1,3) (1,4,3) (2,3,3)... 4 (1,1,4) (1,2,4) (2,1,4) (1,3,4) (2,2,4) (3,1,4) (1,4,4) (2,3,4)... 5 (1,1,5) (1,2,5) (2,1,5) (1,3,5) (2,2,5) (3,1,5) (1,4,5) (2,3,5)... ...
Taking cross sections produces a list that looks like:
(1,1,1) (1,1,2) (1,2,1) (1,1,3) (1,2,2) (2,1,1) (1,1,4) (1,2,3) ...
I create the listing of ordered quartets in the same fashion.
(1,1,1,1) (1,1,1,2) (1,1,2,1) (1,1,1,3)...
Continue the process until you get tired. Ambitious students could write a computer program that creates a listing of sets with n elements. If you want to drop the duplicates, by all means do so. You will notice that the overall size of the sets is smaller. It is same as the difference between combinations and permutations.
To make the power set (well a power set with dups) simply put each of the lists in a table and take cross sections of the table:
(1) (2) (3) (4) (5) (6) (7) (8) ... (1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4) (2,3) ... (1,1,1) (1,1,2) (1,2,1) (1,1,3) (1,2,2) (2,1,1) (1,1,4) (1,2,3) ... (1,1,1,1) (1,1,1,2) (1,1,2,1) (1,1,1,3) ... ...
This listing would look something like:
(1) (1,1) (2) (1,1,1) (1,2) (3) (1,1,1,1) (1,1,2) (2,1) (4) ...
This listing will eventually include all possible finite orderings of sets.
The process of ordering the power set is more clerical than intellectual. As we perform the process, we notice some of the same layering that occurs when we talk about operations on large sets.
We notice that the size of the set of the first n ordered singles is n.
The set of ordered pairs that can be made with the first n integers has 2^n members.
The size of the set of ordered triples that you can create with the first n integers is 3^n and so on through n^n. The size of the set of all ordered collections is: n + 2^n + 3^n + 4^n...
Most treatments of the power set take the extra set of step of removing duplicates. Question: What size would these sets be if you removed the duplicates?
Using the above methodology, it is possible to assign a unique integer index to each and every conceivable collection of integers. The index might be extremely large, but the index of the index is a discrete, finite number.
We can create functions to calculate the index for a particular collection. For the top row of the final table contains a listing of ordered singles. The ordered singles have the positions:
(1) is in position 1
(2) is in position 3
(3) is in position 6
(4) is in position 10 and so on.
We could write an equation to calculate the position of each of the ordered single. Note, the position of the ordered single x is greater than or equal to x. When x is large, the position is substantially greater than x.
A fun classroom exercise is to write equations for finding the position of different orderings of integers. For example, you might write an equation that would find the first n even numbers. Computer programs could write a program that finds the position of a an ordering input by the end user.
We can find a position for fixed length collections. What about infinite collections...like the set of all even numbers?
| Unbounded Sets |
|---|
|
I described the Integers as an "infinite set of finite elements". This
is actually a contradiction of terms. I understand an integer as a finite string
of numbers, but to truly have an infinite collection of integers, wouldn't one
of the integers necessarily be infinite in length?
The contradiction is one of many reasons why I find it best to work with more tangible notions such as arbitrarily large sets and unbounded sets, than to make claims on the completed infinite. |
Integers and rational numbers can be represented by a finite string of decimals. An integer can be represented by a single string of decimals, for example 1234. A rational number can be written as two decimals with a numerator and denominator, for example 1/2 symbolizes one half. Both strings involve decimals and are inherently finite.
Irrational numbers, such as the square root of two, cannot be represented with a finite string of decimals. We can, however, create a formula that produces an endless string of decimals that gets closer and closer to the value of the real value.
I am not an expert on transfinite theory. However, in my brief experience with the subject, I believe that most of the mathematics of the theory can be explored simply by talking about different levels of large sets. It seems to me that it would be possible to build the mathematics without having to lean on the metaphysics or the confusing diagonal method.
When can learn a lot about sets by studying the order of the sets. For example, it is possible to create a list of the positive integers that is in increasing order. In the list (1, 2, 3, 4, 5 ...), the nth item in the list is greater than the n-1 item.
We can order the set of positive halves in the same fashion. (.5, 1, 1.5, 2 ...). Every item in the list is greater than its predecessor.
We can not perform the same feat on the set of all integers. The set of all integers includes both positive and negative numbers. Since we do not know the largest negative number, we do not know where to begin the list. We can order the set of integers with two unknown directions. (...-2, -1, 0, 1, 2...).
The rational numbers provide the same challenge. We cannot write a list of the positive rational numbers in ascending order. It is easy to prove this since there will always be a rational number between any two given rational numbers. create a two dimension matrix where the numbers ascend in order. In the following I start each line with an integer, and show all the unique rationals that can be produced with that number as a denominator:
0 1: 1, 2, 3, 4, 5, ... 2: 1/2, 3/2, 5/2, 7/2, 9/2 ... 3: 1/3, 2/3, 4/3, 5/3, 7/3 ... 4: 1/4, 3/4 5/4, 7/4, 9/4 ... 5: 1/5, 2/5, 3/5, 4/5, 6/5 ... 6: 1/6, 5/6, 7/6, 11/6 13/6 ... 7: 1/7, 2/7, 3/7, 4/7, 5/7 ... ...
We cannot write the list in ascending order, but we can write the numbers in a two dimensional table where each row is in ascending order. What happens if we want to list both the positive and negative rational numbers?
When we look at complex numbers, we find that we need to add more dimensions to our ordering process.
Finally, when we look at the real numbers, we find ourselves dealing with an infinite set of strings that are infinite in length. This involves a higher level of ordering than the rationals.
The fact that different sets seem to have different properties when we order the set indicates corresponds elegantly with the natural layering indicating that these different layers have dramatically different natures and attributes. The ordering of sets adds further credence to the supposition that the study of large sets is a rich area of investigation.
The challenge of the teacher isn't just to drive a dichotomy between rational and reals, but to introduce the student to the fascinating nature of mathematics.
I believe that Rich Theory could be expanded into a more interesting foundation for the study of set theory than the diagonal method used in most texts. I realize that I come from the perspective of a classical liberal arts student peering into the world of mathematics, and have little doubt that greater minds could take the seed of this article and grow it into an even richer discussion of higher mathematics.
Descriptive Mathematics - - Index
- - Sponsors
©2003 Kevin Delaney