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What is אo? Aleph-Null and Transfinite Numbers (July 28, 2024) Summary: The numbers we generally use are "finite" in the sense that they represent limited quantities. Mathematicians, however, sometimes encounter infinitely large entities that cannot be described using finite numbers. Instead, they use "transfinite numbers" which have distinctive, thought-provoking characteristics.
In the early days of personal Web sites, it was common to see a "visitor counter" on the home page of a Web site. The counter showed how many people had visited the site so far. For example, you might see: You are visitor number 5,478 to visit this Web site. Every time someone new visited the site, the number on the counter would increase by 1, like the odometer in a car: You are visitor number 5,479 to visit this Web site. You are visitor number 5,480 to visit this Web site. Today, if you look on the home page of the Harley Hahn Web site, you will see the message: You are visitor number אo to visit this site since 1994. On the home page of Harley Hahn's Guide to Unix and Linux, you will see a similar message: You are person number אo to visit the online edition of Harley Hahn's Guide to Unix and Linux. The purpose of this essay is to help you make sense out of these messages. Specifically, just what is the strange symbol that looks as if it might be some type of bizarre counter? To answer this question, you and I will need to talk about an interesting type of mathematics, one that may be new to you. So before we start, I want to make you two promises: By the time you have finished reading this essay:
• You will understand what you have read.
To begin, you and I need to take a few moments to discuss the difference between numbers (which are ideas) and numerals (which are written symbols).
Let's say I decide to have five groatcakes for dinner. I send email to the local supermarket: "Please set aside 5 groatcakes for me. I will be in to pick them up in 20 minutes." When I think of the idea of "five" (in this case, groatcakes) and the idea of "twenty" (minutes), these two ideas exist in my mind. However, other people can't read my mind so to communicate my ideas, I write them down using the characters "5" and "20". When I do so, I am using symbols to represent my ideas. In mathematics, we define these two concepts precisely: a NUMBER is an idea, an arithmetic value that we use to think about a quantity. A NUMERAL is the symbol or name that we use to represent a number. In other words, we think using numbers; we communicate using numerals. For example:
The numeral 0 represents the idea of the number we call "zero";
What about אo? Actually, the symbol אo is also a numeral. Specifically, it is the numeral that mathematicians use to represent the idea of a transfinite number called "aleph-null":
The numeral 0 represents the idea of the number we call "zero";
The numeral אo represents the transfinite number "aleph-null". So 309 is to "three hundred and nine", as אo is to "aleph-null". Now, all we have to do now is answer the following two questions:
1. What is a transfinite number?
Once we can answer these questions, the symbol אo will make sense. Believe it or not, the answers to these two questions are fascinating. However, before I can explain, you and I will need to discuss several other topics. (I promise you it will be fun.) To start, I want to spend a few minutes talking about "sets".
Mathematical definition: A SET is a collection of distinct, unordered, well-defined things. I realize that (right now) this definition is complicated and somewhat vague, so let's unpack it one word at a time, starting with an example. When we write down a set, we use brace brackets (curly brackets) to enclose the various things that comprise the set. For example, here is a set containing five things: {1 2 3 4 5} In this case, the set contains the numbers 1 2 3 4 5. Notice how we use the left and right brace brackets to contain the set. Within a set, we call the various "things" ELEMENTS or MEMBERS of the set. The set in our example has five elements: 1 2 3 4 5. To make a set description easier to read, we commonly use commas to separate multiple elements: {1, 2, 3, 4, 5} It doesn't matter what in order we put the elements of a set. For example, the following sets are all considered to be the same:
{1, 2, 3, 4, 5}
Here is an even simpler set that has only one element: {3} In mathematics, most sets contain numbers. However, this does not need to be the case. Here are three sets, each of which has six elements. The third set contains the names of various birds.
{6, 5, 4, 3, 2, 1}
The number of elements in a set is called the set's CARDINALITY. For example, the set {1, 2, 3, 4, 5} has a cardinality of 5. In our last example, all three sets all have a cardinality of 6. To be part of a set, the elements within the brace brackets do not need to be numbers. It is only necessary for all the elements to be things that we can think about, and that they are distinct from one another (that is, no duplications). Here is an example of a set that has a cardinality of 8, which means it has 8 members: {chicken, dove, robin, pheasant, blackbird, sparrow, starling, woodpecker} To make it easy to refer to sets we usually give them names. Although a set name can be anything, in mathematics we generally use short, simple names, such as a single uppercase letter. Here, for example, are the three sets shown above, each of which now has it's own name:
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
We can now talk about a specific set by using its name. For example, we can make the following statements:
A, B are C are three sets.
For convenience, we use an abbreviation to indicate cardinality. Most often, we place vertical bars on each side of the name:
|A| = 10 ← means "the cardinality of A is 10"
So, in this case, we could write:
|A| = |B|
Instead of vertical bars, you will sometimes see a small "n" followed by the name of the set in parentheses:
n(A) is the same as |A|
For example:
n(A) = n(B)
To summarize: 1. A set can contain any type of item that it makes sense to think about. 2. All the elements of a set must be distinct from one another (no duplications). 3. The elements of a set are in no particular order. You can now understand our original definition: A set is a collection of distinct, unordered, well-defined elements.
It is often possible to define a set simply by showing all its elements. For example, the following three sets all have a cardinality of 10, which is easy to see because I have specified all the elements for each of the sets:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
If I asked you to describe the three sets, it would be easy. The first set contains the first 10 numbers; the second set has the first 10 even numbers; the third set has the first 10 odd numbers. However, what if we want to define the set of the first 100 numbers; or the first 100 even or the first 100 odd numbers? To do so, we can list the first few numbers, followed by three dots in a row, followed by the last few numbers:
{1, 2, 3, 4, ... 97, 98, 99, 100}
The three dots indicate that the elements of the set continue in the established pattern, and as long as the pattern is obvious we can make our set description even shorter:
{1, 2, 3, ... 100}
Here is an example that doesn't involve numbers: the set of all the letters in the English alphabet: {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z} Again, because the pattern is obvious, we can make our set description shorter and simpler by using three dots: {A, B, C, ... Z} Alternatively, we can describe the set in words without having to show actual symbols or letters (as long as we can do so in a way that is both precise and understandable): "The set of all the letters in the English alphabet" In fact, when we want to describe a set that has a very large number of members, the only way to do so is to use either a precise written description or to show a pattern using three dots:
"The set of all the numbers from 1 to 1,564,598"
Here is a slightly more complex set expressed in three different ways.
"All the multiples of 5 that are less than or equal to 1 trillion"
(Note that 10 raised to the power of 12 is 1 trillion.) Now let me ask you a question: What characteristic do the sets we have discussed so far have in common? The answer is that all these sets have a specific number of elements; that is, a specific cardinality. To show you what I mean, here are some of the sets we have looked at along with their cardinality (on the left):
1 ← {3}
The next example is a bit more complex. How many elements are in the following set? "All the multiples of 5 that are less than or equal to 1 trillion" which is the same as:
{5, 10, 15, 20, ... 1,000,000,000,000}
1 trillion divided by 5 is 200 billion, so there are 200 billion multiples of 5 that are less than or equal to 1 trillion. Thus, the cardinality of this set is 200,000,000,000. Okay, 200 billion is a large number. But with a little imagination, we can describe even larger sets:
• "Number of hairs on the heads of all the people in the world"
Although we will never know the actual number of elements in these last three sets, what we do know is — like all the examples above — they are specific numbers. The mathematical term for this is FINITE, which describes any quantity that has a bounded limit. A set that has a finite number of elements is called a FINITE SET, and, indeed, every set we have talked about so far, including the last three strange sets (hairs, sand, galaxies), has been a finite set. All finite sets have a finite cardinality — an actual number — even if we don't know what that number is. At this point, you might ask: Are there such things as non-finite sets? You betchum, Red Ryder. What set do you think you would get if you started to count at 1 and never stopped? In mathematical terms, the set would look like this: {1, 2, 3, 4, 5, ...} It looks simple, but looks can be deceiving. What we have here is nothing less than an unbounded set that goes on and on forever. And while you are thinking about that, I'll poise another question: What do you think the cardinality (number of elements) is of this unbounded set? I don't want to keep you in suspense, so I'll tell you the answer. The cardinally of the set: {1, 2, 3, 4, 5, ...} is אo which is pronounced "Aleph-Null".
The eminent German mathematician Georg Cantor (1845-1918) was born in Russia, the oldest of six children. At the age of 11, his family moved to Germany where Cantor grew up and was educated. Even as a youth his mathematical skills were considerable and, today, Cantor is remembered as the originator of the areas of mathematics called set theory and transfinite numbers. Indeed, the terminology and the concepts we have talked about in this essay are based on Cantor's early work with sets. At the time, these concepts were considered new and innovative. However, the ideas he developed somewhat later (between 1874 and 1884) proved to be even more avant-garde, significantly ahead of their time. After creating what we now consider to be basic set theory, Cantor asked himself the same types of questions that I posed at the end of the last section: How should we think about sets that have an unbounded number of elements? and, What cardinality could we possibly assign to such sets? Recall the set of all the whole numbers, starting from 1, going on forever: {1, 2, 3, 4, 5, ...} In mathematics, we refer to this set as the NATURAL NUMBERS or the COUNTING NUMBERS (because we use them to count). It is easy to see that the cardinality (number of elements) of any finite set is a natural number. For example, consider the set of all the even numbers less than or equal to 100: {2, 4, 6, ... 96, 98, 100} The cardinality of this set is 50 because, when you count, half the numbers are even. Consider the set of "All the multiples of 5 that are less than or equal to 1 trillion". As we discussed above, the cardinality of this set is 200 billion. Finally, recall the set "All of the stars in all the galaxies in the universe". There is no way to ever know exactly how many members are in this set, but it's safe to say it is an extremely large number Even though 50 is a (relatively) low number; 200 billion is a high number; and the number of stars in the universe is an extremely large number; these three sets have something important in common. The number of elements of each of them, their cardinality, is a natural number. In fact, the cardinality of any finite set is a natural number. But what about the set of natural numbers itself? How can we establish its cardinality? It is impossible to even imagine counting all elements in this set because, by definition, the sequence of all the natural numbers never ends. Cantor's brilliant insight was to consider that the cardinality of any such set to be greater than any existing (natural) number. As such, he needed a brand new number to represent this idea. Cantor named this new number "Aleph-Null", Aleph being the first letter of the Hebrew alphabet. And to represent the number Aleph-Null, he created a numeral consisting of the symbol for the Hebrew letter Aleph, followed by a small subscript zero (which I will explain later): אo (Remember, numerals are the symbols we use to talk about numbers.)
In mathematics, we use the term INFINITELY LARGE or (more simply) INFINITE to refer to anything that is large without limits; that is, unbounded. (In passing, I will mention that we refer to anything that is small without limits as being INFINITELY SMALL. That doesn't concern us here, but it is the fundamental concept upon which calculus is based.) To continue, we refer to any set that contains an infinite number of elements as an INFINITE SET. For example, the set of natural numbers is an infinite set. As I mentioned above, it is the custom to refer to specific sets using a single uppercase letter. Generally, we can use any letter we want, such as: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} The most important infinite sets, however, have all been given standard names: a single uppercase letter printed in what is called a "doublestrike" pattern. The set of natural numbers, being the most important infinite set, has such a name. It is referred to simply as N, and when printed in doublestrike it looks like this: ℕ Thus, we can define the set of natural numbers as follows: ℕ = {1, 2, 3, 4, 5, ...} Are there any other infinite sets? Yes, there are. In fact, there are an infinite number of them. (Let's pause for a moment to think about that.) Here are two more simple infinite sets that will be familiar to you: the set of all the even numbers, and the set of all the odd numbers. For some reason, these two sets are not considered important enough to have a standard name printed in doublestrike so, for now, we'll just call them E and O.
E
= {2, 3, 4, 6, 8, ...}
If you think about this for a bit, it may occur to you to wonder whether on not these three sets are same size. That is, do the Natural Numbers, the Even Numbers, and the Odd Numbers have the same cardinality (number of elements)? At first, it would seem that ℕ should be twice as large as E or O. To be sure, all three sets do contain an infinite number of elements, but perhaps some infinite sets are larger than others. Actually, as strange is that might seem, it is the case that ℕ, E and O are all the same size. That is, there are just as many even or odd numbers as there are natural numbers. This is not something you would expect. Even more unexpected, however, it is also the case that some infinite sets are larger than others. Indeed, there are infinite sets that actually have a larger cardinality (more elements) than ℕ, E and O. We'll talk about some them later, but first I'd like to draw your attention to two more well-known infinite sets that have the same cardinality as the natural numbers: the Integers and the Rational Numbers. The INTEGERS consist all the positive and negative whole numbers, as well as the number zero: {... -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5 ...} Notice how we can use three dots in two different places to indicate an infinite number of elements in both directions. Like the natural numbers, the set of integers has a standard name: the letter Z printed in doublestrike, (The Z comes from the German word zählen which means "to count numbers".) Although all the integers (except 0) are either negative or positive, the plus-signs are optional. Thus, we can define the integers as follows: ℤ = {... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ...} Before we move on, I want to introduce you to one more infinite set, the set contains all possible fractions. As you probably know, a FRACTION is a number that can be expressed as the ratio of two integers: a NUMERATOR (upper number) divided by a DENOMINATOR (lower number). The only rule is that the denominator cannot be zero, because it is meaningless to divide by zero. Here are some examples: 1/2 3/4 99/100 54/890,083 -5/7 -67/68 5 6 -71 -1 0 You might be wondering, why did I mention 5, 6, -71, -1, 0? This is because any positive or negative whole number can be expressed as that number divided by 1; and zero can be expressed as 0 divided by any (non-zero) whole number. So they really are rational numbers, because can all be written as fractions: 1/2 3/4 99/100 54/890,083 -5/7 -67/68 5/1 6/1 -71/1 -1/1 0/1 The standard name given to the rational numbers is the letter Q printed in doublestrike. (The Q stands for "quotient", the result you get when dividing one number by another.) Definition: the set of RATIONAL NUMBERS consists of all the numbers that can be expressed as the ratio of two integers. ℚ = {all numbers n/d, where n and d are integers and d is not 0} Here is a list of all the infinite sets that we have discussed so far:
E = Even Numbers
Why did I go to all the trouble of introducing you to these five infinite sets? It because of the astonishing fact that all these sets have the same cardinality. In other words, believe it or not, each of these five infinite sets has exactly the same number of elements. And that is what is going to lead us to the definition of Aleph-Null. Reminder: Our goal here is for you to understand what it means when you read: You are visitor number אo to visit this site since 1994.
How do you establish the cardinality of a set? That is, how do you determine how many elements there are in a set? With a finite set, you have three choices: you can count, you can calculate, or you can estimate. When a finite set is small enough, it is straightforward to count the number of members of the set. For example, take a look at the following two sets. How many elements do they contain?
{Border Collie, Poodle, Beagle, Maltese, German Shepherd, Dachshund}
All of these sets have six elements, which means that all have a cardinality of 6. How did you figure this out? Simple, you counted. But what does it mean to say that you counted? It means that you matched up each element in the set with the natural numbers, starting with 1, 2, 3, and so on. For example, here are the members of the first set matched up with the natural numbers, starting from 1:
1 → Border Collie
Let's do the same thing for the second set:
1 → 37
And the third set:
1 → 1
In mathematics, we say that you have established a ONE-TO-ONE CORRESPONDENCE (usually written as "1 to 1 correspondence") between the set of natural numbers and the elements of the set you want to count. This means that each element in the first set (in our case, the natural numbers) has one, and only one, relationship with an element of the second set. The idea of creating a 1 to 1 correspondence to find out the cardinality of a set is so important that I will embody it in a definition: COUNTING means to recognize an orderly relationship between the natural numbers and the set of things you want to enumerate. For this reason, the natural numbers are sometimes referred to as the COUNTING NUMBERS. (Hence the old saying: "A man who knows his numbers is a man you can count on".) It is easy to count manually as long as there are not too many elements in the set. However, when there are too many elements, we can sometimes calculate the cardinality of the set by thinking about it carefully. For example, consider the following three sets:
Set 1 = {1, 2, 3, ... 100}
Set 1 = all the natural numbers up to 100
We could write out all these numbers and count them one by one. However, it is a lot better to realize that there are 100 natural numbers from 1 to 100, and that every other one is either an even or odd number. In this way, we can calculate that there are 50 even numbers (100/2) and 50 odd numbers (100/2):
Set 1 has 100 elements
Better yet, remembering that we can use vertical bars to indicate the cardinality of a set, we can write:
|Set 1| = 100
Calculating can work well when there is no possible way to count manually. For example, what is the cardinality of the set of all the numbers less than 2.5 million that are divisible by 7? (2,500,000 / 7) = 357,142 Finally, let's consider the case in which we want to find the cardinality of a finite set, but it is not possible to do so by counting or calculating. In such cases, it can sometimes be possible to estimate. Here is a real-life example. During my time as a graduate student in Computer Science, I was also taking pre-med classes. I remember the following story as told to our class by the organic chemistry teacher. When he was applying for the entry-level job of assistant professor, the Department Chair asked him to find the answer the following question: "How many ping pong balls would it take to fill this office?" Let us rewrite the question slightly: "What is the cardinality of the set of all the ping pong balls it would take to fill this office?" Actually, the organic chemistry teacher never told us how he answered the question, so let's you and I do it together based on the following numbers:
From this we can calculate the following estimates. 1. The volume of the Department Chair's office is:
1.(floor size) x (height of room)
2. A cube with edges equal to diameter of a ping pong ball would be:
2.40 mm x 40 mm x 40 mm
3. The number of ping pong balls that would fit in the office is:
3.(size of office) / (size of one ping pong ball "cube")
Thus, a reasonable estimate of the cardinality of the set of all the ping pong balls it would take to fill the office is 500,000 (half a million). Summary: We can find out the cardinality of a set (the number of its elements) in three different ways: by counting, by calculating, or by estimating. We know this works for finite sets, but what about infinite sets? Is it possible to "count" the number of elements in an infinite set?
What's your guess? Is it possible to count the number of elements in an infinite set? Surprisingly, the answer is yes, if we are willing to make an adjustment to our thinking. To do so, we need to ask ourselves: What do we really mean by "counting", and how can we apply that to infinite sets? This is the same question that the mathematician Georg Cantor asked himself in the 1870s when he was developing what we now call Set Theory. Before Cantor, nobody really understood the idea of sets or knew how to talk about them, let alone being able to study them mathematically. However, once Cantor began to apply his nascent understanding of sets to the activity of counting, he was able to invent the technique that you and I used above. Specifically, it was Cantor who came up with the idea that when we count, what we are actually doing is creating a 1-1 correspondence between the natural numbers and the elements of the set that we are counting. This worked fine when Cantor was counting finite sets. However, how could he count infinite sets with only the natural numbers to use as a basis for measurement? After much work, he came to the conclusion that, yes, it was possible to count an infinite number of elements, but not by using any of the natural numbers (which, after all, are finite). What Cantor needed was an actual infinite number: a number with a name and a value that he could use to measure the cardinality of infinite sets. So he invented such a number, which he named after the first letter of the Hebrew alphabet, Aleph: א For reasons that I will explain later, Cantor included a small subscripted zero to the right of the Aleph character: אo The most common way to pronounce the name of Cantor's numeral is "Aleph-Null", although some people say "Aleph-Zero"; either is correct. Cantor defined Aleph-Null to be the cardinality of the natural numbers. In symbols, this can be written as: |ℕ| = אo So if anyone ever asks you, How many natural numbers are there? the answer is "Aleph-Null". Using this brand new number Cantor was able, for the first time ever, to use a symbol to express the idea of an infinitely large number. This enabled him to talk about infinite cardinality; not only for the set of natural numbers, but for other infinite sets, many of which are even larger. (Eventually, Cantor was even able to show how to do arithmetic with Aleph-Null.) Because Aleph-Null is so different from other numbers, we need a special way to describe it. Without exception, all the regular, everyday numbers that we use represent a finite quantity. As such, we call them FINITE NUMBERS. This includes all the numbers used by you and me; all the numbers used by scientists, engineers, business people; it even includes all the numbers used by computers. Indeed, most people can't even imagine a number that is not finite. However, Aleph-Null represents an infinite quantity. As such, it was clear to Cantor that this new number deserved a category of its own so, in 1985, he coined the term TRANSFINITE NUMBER (trans being the Latin word for "beyond"). Today, you will also see transfinite numbers referred as INFINITE NUMBERS, which is just as appropriate but not as memorable.
To talk about the cardinality of infinite sets, we need to explore Cantor's method. That is, we must try to create a 1-1 correspondence between the set of natural numbers and another infinite set. If we can do so, it shows us that the second set must have the same cardinality as the natural numbers (which is Aleph-Null). Let start with the set of all the even numbers (E) that we talked about earlier. Is there a way to create a 1-1 correspondence between the set of all natural numbers and the set of all even numbers. To do so, we need to find a way to associate every even number with a unique natural number. What would happen if we tried to count all the even numbers? All we need to do is start with the first even number (2), and continue indefinitely. Remember, counting is simply establishing a 1-1 correspondence between the set of natural numbers and the elements of another set. Here is what it looks like for the first six even numbers:
1 → 2
. .
Do you see the problem? In order to count all the even numbers, we would need to continue counting for an infinite amount of time. (And let's face it, very few people manage to live that long.) Think back to how we approached the task of establishing the cardinality of a finite set. There were three ways to do it. First, we could try to count all the elements in the set. If that didn't work, we could look for a way to calculate the number of elements in the set. Finally, if all else failed, we could at least (in some cases) estimate the number of elements. However, none of those methods will work with infinite sets because you can't count, or calculate, or even estimate infinite numbers. (And, besides, right now, the only infinite number we know is Aleph-Null.) To find the cardinality of an infinite set, we need a completely different approach. The easiest way to see if we can establish a 1-1 correspondence between the natural set of numbers and the infinite set. If we can do that, we can say that the cardinality of the second set must be Aleph-Null. But how can we do that if we can't count, calculate, or estimate? One way is to devise some type of procedure or formula that shows us how to use every natural number to create a unique number within the second infinite set. To illustrate this idea, let's take another look at the infinite set of even numbers and ask the question: For every natural number, is there a way to create a 1-1 relationship to a unique even number? The answer is yes. All we need to do is multiply the natural number by 2. Here is what it looks like:
1 → 1x2 → 2
. .
To avoid the problem of the infinite sequence (which would require an infinite amount of time), all we need to do show that our method will work for all the natural numbers. We do so by defining every even number in terms of a unique natural number: Let n be any element of the set of natural numbers. The even number corresponding to n is nx2. Conversely, let e be any element of the set of even numbers. The natural number corresponding to e is simply n/2. In this way, we have shown that there is a 1-1 correspondence between the set of natural numbers and the set of even numbers. And since the cardinality of the natural numbers is Aleph-Null, it follows that the cardinality of the even numbers must also be Aleph-Null. But, I hear you say, doesn't the set of natural numbers have twice the elements than the set of even numbers? The answer is, No, because you can't use finite arithmetic on infinite quantities. (We'll talk about this idea again later.) Let's try the same thing again with the set of odd numbers. To do so, we just need to observe that to create an odd number from a natural number, we multiply the natural number by 2 and then subtract 1.
1 → 1x2-1 → 1
. .
To avoid the problem of the infinite sequence, we need only define the odd numbers in terms of the natural numbers: Let n be any element of the set of natural numbers. The odd number corresponding to n is nx2-1. And for any odd number o, we can find the corresponding natural number by adding 1 to o. and then dividing the result by 2 (the opposite operation). Therefore, we have a 1-1 correspondence between the set of natural numbers and the set of even numbers. What about the cardinality of the other infinite sets that we have discussed, the integers and rational numbers (fractions)? As with the even and odd numbers, there is a straightforward way to show a 1-1 correspondence between the set of natural numbers and both the integers and the rational numbers. The proofs are a bit more complicated, so I won't show them here. But it is the case, so please take my word for it. This means that set of the natural numbers is the same size as the even numbers, the odd numbers, the integers, and the rational numbers. Using the abbreviations we discussed earlier, we can express this idea using mathematical symbols. The abbreviations are:
E = Even Numbers
So remembering that vertical bars indicate cardinality (number of elements), we can write: |ℕ| = |E| = |O| = |ℤ| = |ℚ| = אo Summary: The following five infinite sets:
• Natural Numbers
all have the same number of elements: Aleph-Null, a transfinite number.
So far, we have discussed only one transfinite number: אo (Aleph-Null) As I have mentioned, there are more transfinite numbers and I will show you them in a moment. Before I can do so, however, I need you to understand the idea of "subsets" and "parent sets". When all the members of one set are contained within another set, we say that the first set is a SUBSET, and the second set is a PARENT SET. Here is a simple example. Consider the two finite sets:
A = {1, 2, 3, 4, 5, 6, 7}
Because all the elements of B are contained in set A, we say that B is a subset of A, and A is a parent set of B. The symbol we use to indicate that one set is a subset of another is ⊆ (it looks like a rounded less-than-or-equal symbol). In our example, we can write: B ⊆ A Here is another example (non-numeric): The set of all dogs in the world is a subset of the set of all animals. The set of all animals is a parent set the set of all dogs. Dogs ⊆ Animals Now, consider the question: For any particular set, what are the total number of possible subsets? Let me show you the answer to that question for set B above. If B = {3, 4, 7} is a parent set, there are 8 possible subsets:
1. {}
To make sure you understand this, I need to make a few comments. First, notice that Set #1 contains nothing. A set that contains nothing is called the NULL SET or an EMPTY SET. By definition, the null set is a subset of every set. Why? Because a null set contains no elements, which means that, technically, all (zero) of its elements are contained in every other set. Thus, the null set meets the criteria of being is a subset of all other sets, including the null set itself. (Mathematics is the offspring of logic.} Second, Set #8 contains all the elements of B, which also meets the requirements of a subset. (Set B contains all the elements of Set #8.) In this way, there is a general rule that every set is a subset of itself. The last comment I have to make is actually an observation and — for the most part — you can trust your intuition here: The cardinality of a subset is always less than or equal to the cardinality of a parent set. As an example, here are the cardinalities of all the subsets of Set B:
3, 4, 7|{}| = 0
It is certainly true that the cardinality of any of these subsets is less than or equal (≤) to the cardinality of Set B (which is 3):
3, 4, 7|{}| = 0 ≤ |B|
After all, it only makes sense. Or does it? The reason I said that you can only trust your intuition "for the most part", is that what we have been discussing is true, but only for finite sets. With finite sets, the size of a subset is always less than or equal to the size of a parent set. With infinite sets, this is not always the case. Sometimes, a subset of as infinite set will have the same number of members as the parent set. For example, recall (from the pervious section) that the cardinality of the Natural Numbers (Aleph-Null) is the same as the cardinality of the Even Numbers, Odd Numbers, Integers, and Rational Numbers: |ℕ| = |E| = |O| = |ℤ| = |ℚ| = אo And yet, it is certainly true that some of these infinite sets are subsets and parents sets of one another. For example, the Even Numbers are certainly a subset of the Natural Numbers. Indeed, consider that:
E ⊆ ℕ (Natural Numbers)
O ⊆ ℕ
ℕ ⊆ ℤ
ℤ ⊆ ℚ and yet all these sets have the same number of elements. In plain English, when it comes to infinite sets, something can be a part of something else, and yet, both things can be exactly the same size. Isn't that strange? (Working with finite numbers may be more useful, but playing around with transfinite numbers is a lot more interesting.)
Aleph-Null was the first transfinite number to be recognized by Georg Cantor. Eventually, however, he was able to show that there were infinitely more transfinite numbers. It is time for you to meet them. In order for me to make a proper introduction, we need to start with a question that may seem unrelated: Exactly how many possible subsets can an infinite set have? The answer, of course, is that an infinite set can have an infinite number of subsets. However, what if we want to quantify that amount? To start, we will look for a formula to find the number of possible subsets for any finite set. We will then extend that formula to infinite sets. Consider the set {3, 4, 7} that we discussed in the previous section. Let's say that we want to form a subset. We can consider each member of the set, in turn, and decide whether or not it should be in the subset. In this case:
3 ← Should it be in the subset? Yes or no?
Since each question has two possible answers, the total number of possible combinations is: 2 x 2 x 2 = 23 = 8 If we remember that, for any Natural Number n: 2 x 2 x 2 x 2 ... (n times) = 2n we can easily extend our reasoning: For any finite set S with a cardinality of n, the total number of possible subsets is 2n. Using our example above:
S = {3, 4, 7}
We see that the number of possible subsets is: 2|S| = 23 = 8 which, as we saw in the previous section, is indeed the case. Here is a slightly more complex example: Find the number of possible subsets of the set of all the even numbers that are less than or equal to 20.
S = {2, 4, 6, ... 16, 18, 20}
The total number of possible subsets is: 2|S| = 210 = 1,024 (If you are a computer programmer, you will, no doubt, recognize that number.) We are now ready to do the same type of calculations for infinite sets. But what will we use for numbers? When we quantify — that is, give a numerical value to — anything finite, we use finite numbers. (Indeed, we have the entire set of Natural Numbers, the so-called Counting Numbers, to work with; and if we need more, there are other sets of finite numbers available, such as the Integers, or the Real Numbers or Complex Numbers (which we will talk about later.) However, when we want to quantify something infinite, finite numbers — no matter how large — are not large enough. To count infinite quantities we need infinite numbers. Although the only infinite number we have (so far) is the transfinite number Aleph-Null, it happens to be a good place to start. Here is our motivation. We know that the cardinality of the set of the Natural Numbers is Aleph-Null: אo = |ℕ| What we want to quantify is: How many possible subsets are there consisting of elements taken from the set of Natural Numbers? To create such a formula, we will use the same approach as we did when finding the total number of subsets of finite sets. With infinite sets, it still holds that, for every element in the set, a specific subset either contains that element or does not contain the element. Thus, the total number of subsets of an infinite set is: 2 cardinality of the infinite set For the set of Natural Numbers, we raise 2 to the power of Aleph-Null. So the total number of subsets of the set of Natural Numbers is: 2אo Not only is this a transfinite number, it is a significantly larger transfinite number than Aleph-Null itself. The name Cantor gave to this second transfinite number was "Aleph-1". So we know two things about the second transfinite number: The value of Aleph-1 is 2 to the power of Aleph-Null; and Aleph-1 is larger than Aleph-Null:
א1 = 2אo
So what is Aleph-1? It is the cardinality of the set of all the subsets of the Natural Numbers. Let's go one step further. Consider the set of all the subsets, of the set of all the subsets of the Natural Numbers. What is the cardinality of that set? I'm sure you have guessed: It's 2 to the power of Aleph-1, and we call it Aleph-2. In fact, we can start with the cardinality of the Natural Numbers and continue indefinitely.
אo = |ℕ|
. .
which means that
אo
< א1
< א2
< א3
< א4
. . .
And as strange as it may seem, these are all the transfinite numbers there are. There is nothing in between.
If you have some math experience, here is an optional, somewhat more advanced topic. If you want, you can skip this section. (However, if you do, please make sure to read the next two sections to finish the essay.) In your mathematical travels, you may have come across four infinite sets that we have not yet discussed. First, you will remember the set of Rational Numbers: all the numbers that can be expressed as fractions. There is also the set of IRRATIONAL NUMBERS: all the numbers that cannot be expressed as fractions. I won't go into the details. However, here are two examples of irrational numbers: √ 2 (the square root of 2) π (the special number pi) Next, the set of REAL NUMBERS consists of all the Rational Numbers and all the Irrational Numbers. Third, the set of IMAGINARY NUMBERS consists of a variation of the set of real numbers. For each real number, there is a corresponding imaginary number. The difference is that, when you square a real number (that is, multiply it by itself), the result is always positive. When you square it's corresponding imaginary number, the result is always negative. Finally, the set of COMPLEX NUMBERS consists of all the combinations of one real number and one imaginary number. Here are some fascinating observations about these four infinite sets: 1. They all have the same cardinality, which is larger than Aleph-Null. 2. Their cardinality is Aleph-1. 3. The statement that the cardinality of these sets is Aleph-1 can never be proven completely. OVERALL SUMMARY The cardinality of the following five infinite sets is אo
• Natural Numbers
The cardinality of the following four infinite sets is א1
• Irrational Numbers
However, this cannot be proven.
ARITHMETIC is the branch of mathematics related to the four basic operations: addition, subtraction, multiplication, and division. Some people also include the use of exponents and logarithms, and extracting roots (such as square roots). However, we are interested only the type of arithmetic that can be done between a transfinite number and a finite number, for example: אo + 1 = ? אo – 1 = ? How do we interpret such calculations? Are they even meaningful? To start our search for meaning, consider the following calculations that use only simple finite numbers:
12 apples + 4 apples = ?
If we realize that we are actually dealing with sets of apples, we have no trouble adding apples to apples; or, for that matter, subtracting apples from apples:
12 apples + 4 apples = 16 apples
These two examples make sense and are easy to understand. However, does it make sense to multiply apples? Or divide apples?
12 apples x 4 apples = ? apples
What about taking the square root of an apple? Or an exponent? Or a logarithm? How about something even simpler: subtracting a large number of apples from a smaller number? 12 apples – 24 apples = ? The real answer is, What we mean by arithmetic depends on the type of numbers we are working with. In the case of our apple examples, it makes sense only to add or subtract, and even then, we can't subtract a large number from a small number. (After all, what we are actually doing is counting, and you can't count negative apples.) Thus, when it comes to sets of apples, only addition and (sometimes) subtraction is meaningful. Now let us consider something slightly different:
12 apples + 4 oranges = ?
It seems obvious that we can add apples to apples; and (if the first number is greater than or equal to the second number) we can also subtract apples from apples. But can we add apples to oranges? Or subtract oranges from apples? The answer is, not directly. But what if we change these two calculations so that both items are adding or subtracting the same thing? In other words, is there a way to think about apples and oranges as belonging to the same set? To do so, we need to ask the question, What do apples and oranges have in common? One answer is, they are both fruit:
12 fruit + 4 fruit = 16 fruit
In other words, we can add or subtract two different types of things as long as we can find some way to think of them as being the same. We can then perform the calculations in a way that makes sense. With a bit of ingenuity, we can apply this same idea to arithmetic using a finite number and a transfinite number, even though, at first glance, they would seem to be quite different. Consider the calculation I mentioned above, adding 1 to Aleph-Null: אo + 1 = ? Like the apples and oranges, we are trying to do arithmetic what what seem to be two different things, a transfinite number and a finite number — and it is not at all obvious how to add together two such fundamentally different quantities. However, let us look for a way in which these two numbers have something in common. In the same way that we recognize the number Aleph-Null as being the cardinality of the entire set of Natural Numbers, we can think of the number 1 as being the cardinality of a set than contains exactly one element. In other words, both numbers are cardinalities of specific sets. So, to interpret Aleph-Null + 1, all we need to do is combine the set of Natural Numbers with the another set that contains only one element. We can then ask the question: What is the cardinality of this new, combined set? The answer is now clear. If we add a single element to the set of Natural Numbers, the cardinality is still Aleph-Null. This is because the Natural Numbers is an infinite set, and adding one more element to an infinite set doesn't change its size. More generally, adding (or subtracting) any finite number of elements to (or from) an infinite set does not change the cardinality of that set. So, if n is any finite number: אo + n = אo אo – n = אo For example: אo + 1,000,000,000 = אo אo – 1,000,000,000 = אo However, right now, for us, the most important calculation is: אo + 1 = אo
Let's return to the reason that you and I started to talk about number sets, finite numbers, and transfinite numbers. It was so we could make sense out of the visitor counter on the home page of my Web site: You are visitor number אo to visit this site since 1994. By now, you understand that Aleph-Null (אo) is not a regular number. It is transfinite number and, as such, it is infinitely large. Thus, to say that someone is "visitor number Aleph-Null" for a Web site is actually a mathematical joke: it just isn't possible for a Web site — even one as interesting as mine — to have an infinite number of visitors. Still, I hope you found it interesting to learn about transfinite numbers, and that you feel that your time was well spent. The last thing I want to explain is: Why do I say that the idea of adding 1 to Aleph-Null is so important? אo + 1 = אo You will remember that, at the beginning of this essay, I showed you an example of an ordinary visitor counter on a Web site: You are visitor number 5,478 to visit this Web site. I pointed out that each time someone new visits the site, the number on the counter increases by 1: You are visitor number 5,479 to visit this Web site. You are visitor number 5,480 to visit this Web site. However, no matter how many visitors my Web site gets, the visitor counter will always stay the same: You are visitor number אo to visit this site since 1994. Now you understand why. It's because adding a finite number (such as 1) to a transfinite finite number (such as Aleph-Null), does not change the value of the transfinite number: אo + 1 = אo This is why, no matter how many times you visit my Web site, you will always be visitor number Aleph-Null, which makes you my favorite visitor.
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