Arbitrarily Large Sets

Before launching into an exploration of the infinite. I find that it is more productive to start with infinity's humbler, hard working cousin: the arbitrarily large set.

An arbitrarily large set has a finite number of members. Yet we can always expand it to fit our needs. Of course, there are a few things that arbitrarily large set simply cannot do. They cannot hold every point on the plane. They cannot hold every digit of an infinite decimal expansion. Despite these limitations, they are are good things to have around...especially in the computer age.

Computers just might be a cure for the "disease of transfinite analysis." These strange computing devices sitting on our desks are completely changing our understanding of mathematics. At the turn of the 20th century, people used continuous methods to estimate large discrete events. At the end of the century, people were more apt to use discrete methods to estimate continuous events.

In layman's terms: people used to use clever devices, like slide rulers, to estimate the sum of a large set of numbers. Today, adding a few hundred thousand numbers together is a breeze. Today, people are apt to use brute force calculation to estimate the answer for equations they could have solved with a little algebraic manipulation.

Computers also have had an extremely profound impact on our understanding of logic. Today, people are adept at writing complex logical algorithms (computer programs), and have a good grasp of programming structures like if statements, while and for loops. Business IT departments have pimple-faced geeks happily tapping out complex logical structures that would have made a 12th century scholastic cleric gag.

Like it or not, the computer industry is busily creating a new understanding of mathematics. One of the really profound changes created in this brave new world is that people have become accustomed to working with extremely large sets of data, and are used to working with variables that have a fixed size and upper limits.

For example, the computer I am using right now uses a 32bit integer. This computer uses 32 on off switches to represent an integer. The first bit determines the sign (plus or minus). This means my computer can represent an integer from -2^31 to 2^31-1 (-2147483648 to 2147483647)...wow.

If I perform an operation that goes outside this bound, I might get an error or a wrong calculation (depending on what I am doing.) If I need to work with larger numbers, I simply have to find a way of using more bits to represent my number. This brings up a really stupid joke:

Q: In 1995, why did Bill Gates instruct Microsoft to switch from a 16 to a 32 bit operating system?
A: He received a number out of bound error while trying to balance his checkbook.

The reason I brought up computers is because computers use a concept similar to the arbitrarily large numbers. When you write a computer program, you specify the amount of space needed for a variable. If that amount of space is insufficient, you need to snag a little more room for your variable. Not all operations require that you change your name space. You just have to be attentive to the size of your variables.

Anyway, let's get back to the math.

There are two ways to create a set of even numbers from a given finite set of integers. We could multiple our set of integers by two, or we could simply take out all the odd numbers. Multiplying the first 5 integers {1,2,3,4,5} yields the set {2,4,6,8,10}. Removing the odd numbers produces the set {2,4}.

The first method produces a set of even numbers that is the same size as our set of integers; however it requires that we increase the size of our variable from 1 character to 2. We have to increase the size of our variable. The second method produces a set of evens that is smaller than our initial set of integers.

Neither of the method is superior to the other. However, when we work with a logical mapping we have to be attentive to whether or not we are increasing, or maintaining the size our name space. Let's say we gathered together the first n integers. The set of even numbers contained in this set is floor(n/2). If we multiple all of the integers by 2, we get a set that is the same size as our original set, but require a name space that is twice as large as the original set. Half of the even numbers in or new set did not exist in the original set.

As you can see, when we perform functions within an arbitrarily large set, we have to be attentive as to whether the function demands that we expand the set, or leave it at its current size. If we choose to expand a set, we should be attentive to the maginiturde of the expansion.

Expanding Large Sets

Let's pretend for a moment that we had a set with n members, where n is an incredibly large number. n in a number that is larger than any other number that we will use in our day to day calculations.

Let's say we decide to increase the set by 1 member. Well, we really haven't done much. We increased the set by 1/n which, in our little universe, effectively increases the size of the set by nothing. We could increase the set by say 100 members. Again, 100/n is small number.

Now, lets double the size of the set. Multiplying n by 2 produces a set that is twice the size of the original set. This act of multiplying has a greater impact on the size of the set that adding even really big numbers (big numbers being less than n).

We could multiply this site by other numbers, such 4, 5, 6. Multiplication produces a set that is several times our original. Multiplication is having a much broader affect on our set than addition.

Okay, now lets square our set. The orginal set has n members. The square of the set has n^2 members. This operation has an effect with is greater than multiplying by even a really big number. We can take n to the third power, fourth power, fifth power, and see mighty powerful things happening to the site. This

In our final operatiion, we will take 2^n power. Just like the squaring operation, this produces a new set that is off scale with the original set.

Before we launch into building a theory of large numbers, I wanted to just take a moment to think about how different mathematical operators affect the size of a different sets.

 Addition has very little impact on the size of the set Mulitplication Multiplication in a dramatic yet tame way. Addition has very little impact when compared to multiplication. Exponential Functions The exponential functions rock. They quickly skyrocket the size of the set to astronomical values. Multiplication has little impact compared to exponential functions.

Reducing Arbibrarily Large Sets

Reducing arbitrarily large sets works pretty much as expected. Subtracting a few numbers from the end of the set has a very minor effect on the set. Dividing by two has a stronger effect. As the divisor for your set approaches n (the size of the set) the size of the resultant set will approach 1. Since the integers are not closed under division, you will notice that dividing an arbitrarily large set might leave a remainder.

Exponential functions, like taking the square root of the size of the set will have the same affect on reducing the set as they did on increasing the size of the set.

 Subtraction has very little impact on the size of the set Division Dividing has a much more dramatic effect. Dividing the set by 2 halves the set. However, as the number your are dividing by approaches the size of the set, the size of the product will approach 1. Exponential Functions

While it is difficult to develop an intuitive understanding of the infinite. It is not quite as difficult to develop an intuitive understanding of large sets. In the next essay, we will finally get to some real mathematics. In it, I intend to build a model of the power set of the integers. While I work on the power set I hope you pay attention to how the name space of my variables increase as I include more and more numbers in the set.