1.1.3 - A Definition of The Counting Numbers

1.1.3

A Definition of The Counting Numbers

Intuition Precedes Inspiration

Mathematics does not begin as a fully formalized system. There are always initial assumptions we use to begin our reasoning. Before we can discuss or organize ideas we have to have some first. Where do ideas come from? Intuition. We are first guided by intuition based on our understanding of how things should work. From this we can begin to formalize a system. We can understand this principle as "intuition precedes inspiration".

Before we begin discussing large numbers in the abstract we need to start thinking about numbers intuitively.

In this first article I will establish a definition of number that we can use as the beginning of our investigation into larger and larger numbers...

Discrete Objects

As humans it is ingrained in us to try to understand our own world. It seems that abstraction is also innate. As such, there are things that we can understand about numbers intuitively, even if we lack the language or formalism to explain it.

Where does the concept of numbers come from, specifically the "Counting Numbers"? We define the concept of the counting numbers in relation to the idea of counting "discrete objects".

We can define discrete objects as objects which are difficult to divide into smaller components or as objects which are convenient to think of as a whole. We live in a world full of such discrete objects. People, dogs, cats, cars, homes, chairs, tables, books, clocks, computers, etc. can all be thought of as discrete objects.

An important assumption about discrete objects is that they are relatively permanent. They don't vanish into thin air, split into many discrete objects, and they remain relatively the same in kind. For example, an apple does not become an orange, the house keys never wink out of existence, they are always "somewhere" even if we can't find them, and a chair does not miraculously become a pair of chairs if we look away for an instant and return our gaze.

Granted that living things reproduce, and in this sense a single object can become many, but this isn't really what we mean when we speak of "discrete objects". Remember that the idea of a discrete object is a mental abstraction. It's a way for us to simplify and generalize our understanding of things contained in the world. In fact even at this very early stage there is a certain degree of abstraction going on here. Firstly there aren't really any strict boundaries in the "real world" between discrete objects since they all exist in the same space and are all made of the same fundamental stuff. Also no object is absolutely permanent and without internal change.

So what are we talking about? An idea. We are conceptualizing the actual object, and considering ideal objects.

The ideal object has a single important property:

An ideal object always exists and always remains exactly itself

As we will see, most mathematical "objects" inherently have this property. Real objects aren't exactly like this. An apple remains an apple for quite a while, until someone eats it or it begins to rot. A rock could stay the same for millions of years, so that's pretty close to the ideal, yet wind and rain will eventually erode it so even a rock is not eternal.

If real objects are neither well-defined nor eternal why bother thinking of things in discrete and unchanging units? Why bother to think of things as eternal when this is not the truth?

However we can't really think of things in any other way; our minds seem to be built to think in terms of discrete units. When we imagine a discrete object as being broken down into smaller parts, each part becomes a discrete object in itself. Each of these can of coarse be further broken down into smaller components which are themselves discrete objects. But at every stage we think of things as collections of discrete objects, and discrete objects as containing no parts. Do we ever reach a point at which our mind conceives of no "objects"; does the discrete ever blend into a continuum?

Likewise, we normally think of our identity as a constant, yet are we the same people we were several years ago? How about a year ago, a month, a week, a day, an hour,a minute, a second ago ... can we really perceive a continuum or do we always understand it as a series of discrete changes?

Our very minds are built up of discrete units called neurons. It's conceivable then that we can only think in discrete terms.

This is a very important issue we will return to. However it can not be denied that "objects" or "units" is a useful way to conceptualize our world, if only for the sake of simplicity.

Representations

Once we can conceive of objects in an ideal and abstract sense we can begin to use representations to refer to the original object.

Several millenia ago humans began to think in more abstract ways. When hunters began to paint the animals they hunted on cave walls they were engaging in a type of abstraction. They were "representing" discrete objects...

This painting may have represented an animal recently hunted. In that case the painting would have acted as a symbol of that animal. These kinds of pictorial representations are the first stage in developing a symbolic language. It is very likely that our letters and number symbols began as pictorial representations of objects that got repeatedly simplified for the purposes of brevity...

When early peoples drew stick-figures did they really think people looked like that, or were they just bad artists? Neither. They were simply using a symbolic language to make it easier to communicate ideas.

Consider this: no matter how sophisticated a realistic painting is, it is still just a duplicate of the original. It can never be the original object. The amount of effort required to create a realistic work is also extremely time consuming. If you wanted to convey the simple idea of a person, you would want to do it with the minimum amount of effort to convey the idea. This is probably the principle by which pictorial representations eventually got simplified into symbolic representations.

Mathematics abounds in symbolism. Even at this early stage we can begin using symbols in our discussion. Just as ancient people might use a painting to represent a beast, we can use letters to represent discrete objects. For example I could use the symbol :

... to represent "a cat". We can say that "C" is "a cat", and can use the equal sign, " = ", to represent this relation between symbol and corresponding object ...

For the purposes of our discussion we will use capital letters to represent discrete objects. For example using "A" for apple, "O" for orange, "P" for pear, etc. ( generally I'll be using the first letter in the name of the object to generate such representations.)

The first important thing to note about objects and their corresponding symbol is that we must be consistent about their usage.

Although it is certainly reasonable to use more than a single symbol to refer to the same object, it would be disadvantageous to use the same symbol for several unrelated objects. If I use "C" to represent a cat, then later use it to represent a dog, then I have left it's meaning ambiguous. Now if I use the symbol again without first specifying what I mean there will be ambiguity present (assuming no contextual clues).

On the other hand it is entirely reasonable for me to use C for cat, and also D for cat, as long as I consistently use the letters C and D to always represent the same object.

Therefore for each and every symbol we create we must assign a single non-ambiguous meaning to it. This is very important because mathematical reasoning is built upon this simple kind of consistency.

If our symbols refer to a single concept, and a single concept only, then we can also say that any symbol is always equivalent to itself. That is ...

C = C

This idea may seem so obvious that it doesn't even need stating. However, it is worth noting as it does serve as the foundation of all later ideas. Furthermore ordinary language is rarely so strict. A single word can have several unrelated definitions, which are meant to be sorted by context. Mathematics on the other hand does not allow us to be so vague. The pay off however will be the ability to make sweeping mathematical generalizations based on the consistency of our reasoning.

This principle of self-equivalence (law of identity), is considered a classic law of thought. It is usually attributed to the works of Aristotle ( classic greek philosopher ), however there is some contraversy over it's true origin[1]. Lyle Burkhead claims that the law of identity originates sometime in the middle ages and could not have originated before 1274. Assuming that is true, does that mean people were confused about self-equivalence before 1274. Of coarse not! It just slipped everyone's notice because it was so fundamental, or it was just too trivial to be mentioned. This provides compelling evidence that intuition indeed precedes inspiration, sometimes by centuries! We may use ideas intuitively before we are even aware we are using them at all!

So to summarize, we have these key properties of Discrete Objects:

They are eternal and unchanging

They are always equal to themselves

Every discrete object must represent a single and only a single real object.

Once we can talk about discrete objects in abstract terms we can begin talking about collections of discrete objects. As we will see, understanding collections is the key to our intuitive understanding of the properties of the counting numbers...

Collections of Discrete Objects

We can now consider discrete objects in tandem rather than isolation. That is, we will now consider objects as being part of "collections" of objects. For example, take a room that contains several chairs. You can consider each chair individually, or you can consider all the chairs in the room as representing a kind of collection. The choice of what kind of collection to consider is our own, but once we have decided there are certain immutable truths about this collection.

First of all, regardless of how the chairs are arranged in the room, there is always the same collection of chairs. What is it that is remaining unchanged about the collection of chairs? Clearly the positions are changing, but the objects themselves remain unchanged, and therefore the collection remains unchanged.

We can demonstrate this symbolically by using "spheres" to represent objects which are part of such a collection. If we rearrange these spheres the collection remains equivalent :

This can be seen as an extension of the "law of Identity". We know any discrete object is equal to itself...

Now consider this. Say "C" is any collection of objects. Let "O" be any discrete object. If a pair of collections are identical and we add the same discrete object to both then the resulting collections should still be equivalent. Now consider a single object as a collection containing only that object. It follows from these statements that a pair of objects must be equivalent to itself ...

Now let's use "O" to represent the spheres. We can then say that ...

O = O implies O O = O O

Furthermore, this can be extrapolated indefinitely to show that any collection is equivalent to itself. If a pair of collections is equivalent and yet another object is included in both then ...

O O = O O implies O O O = O O O

then ...

O O O = O O O implies O O O O = O O O O

...

etc.

Because this notion only considers the issue of inclusion, it doesn't matter in what way we add the objects to the set. Therefore every possible arrangement of the objects must be equivalent. We can also think of a collection as an extension on the notion of wholeness. A collection is made up of discrete parts, yet it can also be thought of as a whole. That whole will not change as long as all it's parts are present and unchanged.

It should be noted that some "collections" require their parts to be in a specific arrangement in order to remain themselves. For example, consider a clock as a collection of gears. Each of it's gears must be exactly where it is in order for the whole to be considered a functioning clock. If the position of those gears are changed, the clock becomes instead a pile of gears. However we rarely consider "clocks" as "collections of gears". Instead we treat it as a whole. The notion of "collection" we are talking about is much more general. It doesn't require that the objects within the collection perform any function. Because of this order is not important.

To better see this we can use objects of different types. I will use "A" for apple, "O" for orange, and "P" for pear.

Imagine a collection of an apple, an orange, and a pear ...

Intuitively we understand that the collection remains the same regardless of the order in which its objects are included. We can say that the collection of an apple,orange, and pear is the same as a collection of an orange,pear, and apple...

In fact we know there are lots of orders in which we can list these objects. When we state the first object we have a choice of either an apple, an orange, or a pear. When we want to state the next object we have only a choice of the pair of objects we didn't chose at first. For the last object we only have the choice of the object we didn't chose in the first or following step. Using our symbols we can list out all of the combinations ...

A O P

A P O

O A P

O P A

P A O

P O A

Any pair of these will be equivalent to each other. This property of collections, that the order of it's objects does not change the collection, we will call the "Principle of equivalent orderings".

Although it seems we may simply have to accept this principle as inherently true without proof, we can actually construct the principle as long as we adopt a few smaller principles. For clarity sake we will need to distinguish between two collections being equivalent and identical. Collections are "identical" when they have the same objects in the same order. Two collections are "equivalent" intuitively when they contain the same objects regardless of order. The following "rules" are intended to establish the equality of two collections "formally" rather than "intuitively".

Extension of the law of Identity: The first principle we need to adopt is an extension of the law of Identity. If an object is always equal to itself, any collection should be equal to itself provided it contains the same members in the same order. This principle is easy to accept, and would probably be accepted implicitly even if I did not state it as such.

Transitive property of equality: Our next principle states that if "A" , "B", and "C" are all collections then " A = B " and " B = C " means that " A = C ".

Reflexive property of equality: This principle states that if "A" and "B" are collections and it is given that " A = B " then it is also true that " B = A ".

Law of Inclusion: Our last principle is also the most important. Let "A" and "B" be a pair of "identical" collections (same objects in same order). Now take an object "X" and include it anywhere within A and anywhere within B to form new collections. The new collections will be equivalent. To understand this think about an identical pair of collections. We can symbolize it as a string of O's. Now we can take X and place it anywhere in each collection. The two new collections will be equal. For example " X O O O O = O O X O O " based on this principle. The "X" can be placed at either end or it can be inserted between any two consecutive objects.

Lastly we will agree that any single object can be considered as a collection containing only that object.

Hopefully these principles are easier to accept than the principle of equivalent orderings. We now want to show that these principles can be used to show the equivalence of all orderings of any collection (prove the principle of equivalent orderings).

To show that this can be used to prove the equivalence of orderings we begin by stating that...

A = A

O = O

P = P

Since every single object is a collection by definition and there is only a single ordering of such a collection then by extension of the law of Identity, every collection containing a single object is equal to itself.

Now take the collection " A " and add " O ". According to the law of inclusion we can insert " O " either to the left or the right and get the same result because " O = O " (To make this easy to see I will color the object which is being moved red). Thus ...

O A = A O

Next we can add " P " to our new collections. Take " O A ". We can insert P to the left,right, or in the middle. In each case the result will be the same . This means that...

P O A = O P A

By the same token we can say that...

O P A = O A P

The transitive property tells us that...

P O A = O P A

and

O P A = O A P

then it follows that ...

P O A = O A P

Now consider the reflexive property...

P O A = O P A means that O P A = P O A

O P A = O A P means that O A P = O P A

P O A = O A P means that O A P = P O A

when all of the above statements are re-organized they show that any of the collections " P O A ", " O P A ", and " O A P " are equivalent to each other...

P O A = O P A or O A P

O P A = P O A or O A P

O A P = P O A or O P A

To simplify this we can use "equivalency chains". These are expressions that use more than a single equal sign. In such an expression every collection will be equivalent to another...

P O A = O P A = O A P

This is an achievement, but unfortunately there are other orderings we haven't covered yet. To get those other orderings we begin with " A O " instead of " O A ". Using the law of inclusion, and the transitive and reflexive properties of equality we can show that ...

P A O = A P O = A O P

... just as we did with " O A ". The real problem is how do we show that the first set of collections is equivalent to the next? In order to do that we need an additional pair. We will begin with " A P " instead. Adding " O " to the pair we can show that ...

O A P = A O P = A P O

Interestingly this shows that...

O A P = A O P

That is, it establishes an equality between a collection from each of the sets. Using the transitive property we can eventually show that all of the orderings are equivalent...

P O A = O P A = O A P = A O P = A P O = P A O

Note that this doesn't just establish the equivalency of orderings for the collection of " A O P ". Since A , O , and P could stand for any collection of unique objects this will be true of anything we choose to substitute it with. For example we could use "C" for cat, "D" for dog and "M" for mouse, and all the possible orderings of " C D M " would be equivalent.

Continuing this logic we can eventually show that all orderings are equivalent regardless of the collection considered.

The proof of this is fairly straightforward. Begin by imagining any collection "C" which has a specific order. Let X be some object occurring within the sequence of C and let O represent the rest of the objects. Then any collection can be written as ...

OOO ... OOXOO ... OOO

... where the location of X is not specified and we don't state exactly how many O's there are. It follows that ...

OOO ... OOXOO ... OOO = XOOO ... OOOO ... OOO

To see this consider the sequence of O's to be a collection. In this case, X, by the law of inclusion can be placed anywhere without effecting the newly created collection. Therefore, regardless of the position of X we can move it to the far left of the list. Next replace one of the O's with Y. Say Y occurs at some position other than the far left. Again we can move Y in the position following X ...

XOOO ... OOYO ... OOO = XYOOO ... OOO ... OOO

We can continue this repeatedly replacing O with another symbol, and moving it to the position directly after the last object moved until every object has been placed along the sequence. Since all of our selected objects are unspecified it means any term can be to the far left, followed by any object in the next position, and so on. This means that our original order would be equivalent to any concievable order via the transitive property of equality. Furthermore the transitive property can then be used to show that any ordering is equivalent to another ordering, and this will be true regardless of the collection considered.

Another way to look at the law of inclusion is that it allows us to move any object to another position in the list without effecting the collection. This means we are free to rearrange objects without effecting equivalency.

An even weaker version of the law of inclusion can be used to prove the principle of equivalent orderings. We only need to accept that we are allowed to flip the order of a pair of consecutive objects without effecting the collection. This is certainly true if we already accept the stronger version of inclusion, because this is equivalent to moving an object from one side of an adjacent object to the other. If we can switch the order of consecutive objects we can eventually move any object to any position without effecting the rest. Thus we would still be able to prove the principle of equivalent orderings even with this modest first assumption.

We now have a basic understanding of what is meant mathematically by a discrete object and a collection. With the inclusion of a single additional idea we can define number...

Object-to-Object Correspondence

We have established when a pair of collections are equivalent, namely when both contain exactly the same objects. However what can we say about a collection of apples and a collection of oranges? They definitely can't be equivalent because they do not contain the same objects. In fact there is no object common to both collections. In this case we say that the two collections are not equivalent. We can use the inequality sign , " =! ", to state this ...

A A A =! O O O

In other words any collection containing only apples is always unequal to a collection containing only oranges. What about these collections ...

A A A O and A A A A

Even though they are nearly identical they are not equivalent. This is because the first collection contains an orange, and the other does not. So we say...

A A A O =! A A A A

(note that non-equivalence is reflexive just like equivalence is.)

So how do we define non-equivalence? The simplest definition is to say that two collections are non-equivalent provided there is at least a single object which is not contained in both collections. Consider that statement for a moment to verify that that simple property defines non-equivalence. It basically means a pair of collections are non-equivalent as long as there is something different about each.

Equivalence requires a greater condition. In order for two collections to be equivalent, for every object contained in one collection there must be a corresponding object in the other.

Earlier we established that any ordering of a given collection is equivalent to another ordering (principle of equivalent orderings). However this does not help establish an equivalency between a pair of arbitrarily given collections. For example ...

A O A A P O A P O P A O P A P A A O P

O A P P A O O O A P A P P O A P A P A

Recall that just because the "lists" aren't identical does not automatically mean they are non-equivalent. These may simply be a pair of orderings of the same collection, but then again they may not. How can we establish their equivalency or non-equivalency in this case?

As the definition above states we must show that for every object in a given collection there is an corresponding (identical) object in the other.

To make this process easier, we will consider objects in the first collection from left to right. If we find a corresponding object in the other collection we will highlight both in red and then subsequently remove them both. At the end, if the pair of collections are in fact equivalent, then every object will be removed from both collections. If they aren't equivalent than at some point we will be able to show that there is an object which is not contained in both collections. Let's begin ...

A O A A P O A P O P A O P A P A A O P

O A P P A O O O A P A P P O A P A P A

now we remove the pair of apples from the collections...

O A A P O A P O P A O P A P A A O P

O P P A O O O A P A P P O A P A P A

now we remove the pair of oranges...

A A P O A P O P A O P A P A A O P

P P A O O O A P A P P O A P A P A

next the pair of apples...

A P O A P O P A O P A P A A O P

P P O O O A P A P P O A P A P A

next the pair of apples (notice the lists are becoming shorter)...

P O A P O P A O P A P A A O P

P P O O O P A P P O A P A P A

remove the pair of pears...

O A P O P A O P A P A A O P

P O O O P A P P O A P A P A

remove the pair of oranges...

A P O P A O P A P A A O P

P O O P A P P O A P A P A

remove the apples...

P O P A O P A P A A O P

P O O P P P O A P A P A

remove the pears...

O P A O P A P A A O P

O O P P P O A P A P A

remove oranges...

P A O P A P A A O P

O P P P O A P A P A

you get the idea, let's continue until we come to a decisive conclusion...

A O P A P A A O P

O P P O A P A P A

O P A P A A O P

O P P O P A P A

P A P A A O P

P P O P A P A

A P A A O P

P O P A P A

P A A O P

P O P P A

A A O P

O P P A

At this point you should be able to see that we are about to run into a snag ...

A O P

O P P

There is no corresponding apple in the other collection. This proves that the two original collections are non-equivalent. Thus ...

A O A A P O A P O P A O P A P A A O P =! O A P P A O O O A P A P P O A P A P A

You may argue that this method is tedious and there are better ways to determine whether or not a pair of lists are equivalent. However the more efficient methods usually involve counting which is in turn based on number, but we have not yet defined number, so we can not use those methods here.

The idea we have used here is the idea of "object-to-object correspondence". Basically an object-to-object correspondence establishes a "link" from every object in a collection to a unique object in another.

Object-to-object correspondence is the real foundation of the number concept. I will now define numbers in a simple an logical way...

A Definition of Number

Let's return to the collection of apples and the collection of oranges used earlier. Consider these collections ...

A A A

O O O

Even though we know that they are non-equivalent because there is no corresponding apple in the collection of oranges and vice versa, we can still establish a object-to-object correspondence of sorts.

To do this I will use the colors red, green, and blue to highlight objects within the collections. Now observe this...

A A A

O O O

Note that we can establish an object-to-object correspondence between objects of the same color. The red apple will be paired with the red orange, the green apple with the green orange, and the blue apple with the blue orange. How is it we can establish this kind of object-to-object correspondence if the collections are clearly non-equivalent?

To show collections are equivalent we need to establish a special object-to-object correspondence that matches identical objects, but the concept of object-to-object correspondence is more general than this. Object-to-object correspondence can be used to link different objects. Here we are linking each apple to a corresponding orange.

When such an object-to-object correspondence can be established between a pair of collections such that for every object in some collection we can match it up with a unique (but not necessarily identical) object in the other collection and vice versa, we say that the pair of collections "contain the same number of objects". If such an object-to-object correspondence can not be established we say the pair of collections "do not contain the same number of objects".

This notion can be used to define number. To make it viable we say that every collection of discrete objects defines a number which represents "the number of objects in the collection".

To understand this symbolically we can place paratheses around any collection, and this expression will represent the number defined by the number of objects in the collection.

So we can say that ...

( A A A )

...represents a number. Another way to describe the concept of number is as the "size of a collection". Note that these definitions are necessarily dependent upon language and an intuitive grasp of it's meaning. In this way we can say the foundation of mathematics is our own intuition.

Given these definitions we can now establish some interesting truths. We can say that even though...

A A A =! O O O

we can show that ...

( A A A ) = ( O O O )

...because a object-to-object correspondence can be established between the collections, and by definition this means the collection contain the "same number" of objects. We will say that the "size of each collection" is equivalent in this case. We will also say that the two collections are "similiar".

We can also establish when the size of collections are non-equivalent, or dissimiliar. They are dissimiliar when they "do not contain the same number of objects". To show this we must establish that there DOES NOT exist any object-to-object correspondence between the pair of collections.

Let's use an easy example. Lets establish that the collection " A O " does not contain the same number of objects as the collection " A O P ". To do this we must show that no object-to-object correspondence exists.

Every correspondence relation will relate a single unique object from a collection to a unique object in the other.

Take " A O ". The apple and orange can be related to a single unique object in the other collection. Say we relate them in this way...

A O

A O P

If we pair of the colored objects we have established an object-to-object correspondence between the pair of collections, so they must be equivalent, right? Well clearly that is false. If you read the above definition for "contains the same number" however you will see that we must be able to establish such an object-to-object correspondence beginning with either collection, not just a single collection.

So we begin with " A O P " this time. Now we must establish an object-to-object correspondence where each object in the collection has a unique corresponding object in the other.

So we begin with the apple in " A O P ". It can correspond to either the apple or orange in the other collection. So let's establish a correspondence between the apples for argument sake...

A O P

A O

Next we consider the orange in " A O P ". It can correspond with the orange which is distinct from the apple in the collection " A O " ...

A O P

A O

Now we consider the pear in " A O P ". The problem is that if we correspond it with either the apple or orange in " A O " it is not a distinct correspondence, thus this can not be an object-to-object correspondence.

Just because this correspondence didn't work doesn't mean we should assume its impossible. To show that no object-to-object correspondence exists we need to show that every possible correspondence fails to establish an object-to-object correspondence.

We can also consider relating the apple to the orange and the orange to the apple...

A O P

A O

but again the issue is that the pear then has no unique object to be related to. For arguments sake we might say the problem is the "exclusion" of the pear. So let's begin by relating the pear to something, say the apple...

A O P

A O

Next we can relate the orange to the orange ...

A O P

A O

but now there is no distinct object "left over" for the apple, so the problem remains. Obviously if we switch the order of correspondence the apple will still have nothing left. What if we set up a correspondence for the apple and pear? ...

A O P

A O

Then the orange has no distinct object available, regardless of how we relate the apple and pear to the objects in the other collection. We have exhausted the possible correspondences (we can ignore the order in which the correspondences are established, as we can switch the order of correspondences and still have the same total correspondence). Thus we can say that there DOES NOT exist an object-to-object correspondence between the collections. This means the size of the collections is non-equivalent and we can state this as...

( A O P ) =! ( A O )

You may wonder why I had to discredit every possible correspondence. Just the fact that a single correspondence doesn't work means all shouldn't work, no? There is a reason I defined number this way. For the collections we are considering here it is true that if we can not establish a correspondence than we will not be able to establish any other correspondence. However we will later see that a more general class of collections defies this basic intuition. There are collections where you can both establish an object-to-object correspondence, and yet show that you can establish a correspondence where there are still "objects left over".

For now however we will use the common sense notion of non-equivalency of collection size. That is, if we can show that a correspondence leads to objects being "unpaired" in either of the collections we will say that the size of the collections is non-equivalent.

This makes our job considerably easier mind you. Lets say we consider the collections...

A P O O A

A O P P O A

To establish that the pair of collections are different sizes using the above method would be very time consuming because of the sheer number of possible correspondences. Using the new definition however, we simply have to establish a correspondence and show that some objects are left unpaired.

To establish this we will remove paired objects. We will highlight pairs in red. This time we needn't make sure that the pair of objects is identical...

A P O O A

A O P P O A

P O O A

O P P O A

O O A

P P O A

O A

P O A

O A

and finally we have ...

because there is an apple left over this means that a correspondence does not exist and the size of the collections is not the same. Thus...

( A P O O A ) =! ( A O P P O A )

Notice this interesting relation between the sizes of collections and the equivalency of collections. If a pair of collections is the same size it doesn't follow they are the same collection. We saw this in the example with the collection of apples and the collection of oranges. However what can we say about a pair of collections being equivalent or not if their sizes are different.

Common sense should inform you that if the sizes are different than by neccessity they are different collections. Recall that we establish equivalency of collections by a more stringent object-to-object correspondence of relating identical objects. If we can not even establish a correspondence between general objects it follows we can not find a correspondence relating identical objects. Thus we can make these important observations...

If A and B are collections and (A) =! (B) it follows that A =! B

furthermore if A = B it follows that (A) = (B)

So far we have simply formalized intuitive notions of collections, objects, and numbers. Now we can use all of this to begin exploring the counting numbers...

Numbers without names

We define the counting numbers as the collection of all numbers that can be defined via the size of collections of discrete objects. Examples of the counting numbers are thus...

( A ) , ( A O ) , ( A O P ) , ...

Note that we have defined the counting numbers without having any names for the numbers ! In fact we haven't even established any notation for them. Note that we can use (AAA), (OOO), or (PPP) to define the same number. We don't have unique designators for each number. It is therefore possible to define number without appealing to a definite notation. Interestingly using the paratheses allows us to describe numbers quite easily.

We can now begin our discussion of "large numbers"

Size Order

We will now define what a "large number" is. A large number is simply a number which is "larger", "greater", or "more" than another number. Of coarse I have to define what I mean by "larger" in this case.

We will say that when we pair off objects between a pair of collections and there are objects left over, the collection with the left over objects is the larger collection, and the collection with none left is the smaller.

Take for example " A A " and " A A A ". We pair them off and ...

A A

A A A

Since " A A A " still contains an unpaired apple we say its a larger size than " A A " which has no unpaired apples. We will use the " < " symbol to mean that the number on the left is "smaller" than the number on the right, and the " > " symbol to mean that the number on the left is "larger" than the number on the right. Thus we say that...

( A A ) < ( A A A )

( A A A ) > ( A A )

These state that " A A " is smaller than " A A A " and that " A A A " is larger than " A A ".

Using these definitions we can establish an order to the counting numbers. We list them from the smallest to the largest.

Is there a smallest counting number. Yes. Note that (A) represents the smallest counting number. In fact any collection containing a single discrete object will be the smallest counting number. Any other counting number is by definition greater. thus (AA) > (A) and (AAA) > (A) and so on.

We can also establish the "next" number for any counting number. Take the number (A). To get the next number simply include an additional object in the collection. We can say that (AA) is the next counting number after (A). After (AA) would come (AAA) and then (AAAA) and so on.

So now we can discuss the order of sizes. The counting numbers can be listed from smallest to largest starting with (A), then continuing with (AA), then (AAA), and so on as far as you'd like.

Note that the signs " < " and " > " , known as inequality signs, have a reflexive property. That is...

If A and B are collections and (A) < (B) then (B) > (A)

If A and B are collections and (A) > (B) then (B) < (A)

These signs are also transitive, just like the equal sign...

Let A, B and C be collections. If (A) < (B) and (B) < (C) then (A) < (C)

Because of this simple property we can confidently say that if a number (A) is larger than another number (B), it is also larger than any number smaller than (B).

We will say a number is "very large" if it is not only larger than another number, but larger than "a lot" of numbers. The use of "a lot" is necessarily subjective. We will be considering many kinds of large numbers, some larger than others.

Of all the numbers we will consider only (A) can not be said to be "large" in the above sense. In order to be large it needs to be larger than another number, but since (A) is the smallest number it is not larger than ANY counting number. We will then say that (A) is the only truly "small" number. All other counting numbers are therefore "large"

The number (AA) is large, but we will say it is not "very large" because it is only larger than a single other number, namely (A).

The number (AAA) is the first number we can say is "very large" because it is larger than both (A) and (AA). That is, it is larger than more than a single number.

Obviously any number larger than (AAA) must also be "very large" according to all the definitions we have established.

We now have everything in place to ask the question that will serve as the impetus for this entire website...

How many counting numbers are there?

Now that we have the concept of numbers it's natural to turn the question of numbers on itself. If we can ask how many objects there are, we can also ask "how many numbers". Let's say we allow the use of numbers as objects. We can then "count" numbers.

For example if we consider the numbers (A) and (AA) we can say we have (AA) numbers. We can say...

( (A) (AA) ) = (AA)

If we consider the first (AAA) numbers we have (A), (AA), and (AAA). We can say ...

( (A) (AA) (AAA) ) = (AAA)

Here we are only considering some of the numbers we can define. We can of coarse define more by considering larger and larger collections. What we want to know is "what is the size of the collection of ALL counting numbers".

The problem seems to be that no matter what collection of counting numbers we consider we still haven't considered ALL counting numbers. Take this collection of numbers...

(A) (AA) (AAA) (AAAA)

To ensure that there is a number we haven't considered, take the largest number in the collection, namely (AAAA). Now add another object and we know by definition it will be larger than (AAAA) and by proxy larger than any number less than (AAAA). Therefore it is not equal to any number in the collection and is a number we have not yet considered. So lets say we include this new number...

(A) (AA) (AAA) (AAAA) (AAAAA)

But we can use the same trick to come up with another number which hasn't been considered yet ! In fact for any collection of numbers we can use the trick to come up with another number.

Actually we should have anticipated this. I never put any restriction on the size of the collections we could consider ! Note that we can always define a larger collection just by adding more objects. Furthermore we can always add more objects. Now in the real world this may not be true. There are going to be only so many apples, oranges, pears, people, cats, dogs, etc. in the world, so we can not simply keep adding objects in the real world. But we are now speaking abstractly about collections. Is there a largest collection? Apparently, no.

How can that be? We assume that if we have a collection of ordered objects we must have a first and a last. How can we not have a last number? Because we can always define another larger counting number.

We therefore come to a strange truth. We can not use the concept of the counting numbers to count the number of the counting numbers. We can not number the number of numbers!

Another way of stating this is that no number represents the number of all counting numbers, if such a concept is a number at all !

Does this mean that the number of counting numbers is larger than any single counting number? Let's examine that idea via the notions we have already established.

Any collection we can write of counting numbers does not contain every counting number, because we can generate another counting number outside of the collection. Now say "C" is any such collection of counting numbers. Let "N" represent the collection of all counting numbers (we will assume such a thing is possible even though we can not write it out).

Now we normally show a collection is larger than another by establishing that there will be objects left over in the larger collection.

Now say we pair off numbers in C with the same number in N. If we did this, by necessity every number in C would be paired with a number in N. Therefore there would be "no unpaired numbers in C". Yet N by necessity would still have unpaired objects. We can therefore say according to the definitions we have established that ...

for any collection of numbers C. It follows that the number of numbers is larger than any given number !

Now we can ask, is (N) even a number? It is larger than any counting number, but we have to conclude that it itself can not be a counting number. If it was equal to a counting number we could generate a larger counting number and then (N) would be smaller than (N) , but this is a contradiction because we know N = N implies (N) = (N). (N) therefore exists outside the collection of counting numbers ! Another way to state this is that (N) can not be contained in the collection of N.

Yet (N) must be contained in N, because we defined natural numbers as defined by any collection. N is clearly a collection and therefore (N) must define a counting number.

Thus we have a paradox. (N) is both a counting number and not a counting number.

Savvy readers will of coarse recognize that (N) represents infinity, or the cardinality of the counting numbers, and is not a number in the usual sense. It therefore shouldn't even be included in the same collection. Furthermore all counting numbers are "finite", and so (N) can not be contained in the collection N because it is infinite.

Essentially professional mathematicians can avoid this simple paradox because they place the cardinality of N in another class of mathematical objects called "infinite sets". However I believe the paradox is a valid one, and this really brings home the problems with considering the infinite. Making a distinction between the finite and infinite may help to eliminate the paradox, but unfortunately there is no natural distinction we can make between them (It is of coarse possible to continue the logic here to define properties of the infinite and then use these properties to distinguish it from the finite, but these properties are not readily apparent at first).

The paradox occurs because I didn't place any restriction on the notion of a collection. It was assumed to be well understood. If we can accept that a collection defines a "number", and that the collection of all numbers is a collection then the paradox stands that (N) is both a number and not.

I believe this paradox is the impetus for the notion of the infinite as something separate and different from the finite. It necessitates the distinction, however the paradox still remains given the logic we established. The problem is the logic is easy to accept, though the conclusion is contradictory.

What if (N) doesn't exist? If we can not write out all the members of N how can we be sure that such a collection exists? In short, what reason do we have to believe in the infinite? It is not necessary to consider the collection N in order to define numbers and establish a size order for them. In fact the question "how many counting numbers are there?" can be treated as completely theoretical. Why then have mathematicians made such extensive use of it? It is a concept that provides some clarity and completeness to the notion of the open-ended. The crux of the paradoxical features of the infinite I believe stem from holding the ideas of "all counting numbers" while also claiming there is no "last counting number"

Let's say we define all counting numbers as "finite", and the size of the collection N as "infinite" (meaning simply not finite).

How is it that we can have an infinite number of finite numbers? If N contains only finite numbers how is it itself infinite?

We will be returning to this idea of the infinite and the finite before the end of this web-book. For now we will accept the conclusion that (N) is not a counting number, yet is larger than any counting number.

Bridging the gap between the finite and the infinite

If (N) is larger than any counting number (C) then there must be numbers that are larger than (C) and yet smaller than (N). This follows from the idea that we can add objects to collection C and still have a finite counting number. By definition this number must be smaller than (N) yet larger than (C). Is there some way we can bridge the finite and the infinite? It seems reasonable that if (N) is larger than (C) there must be someway to reach (N) by adding enough objects. At least this seemed reasonable to me as a kid. Yet I also understood that it would be impossible to reach "infinity" or (N) in this case. So I was presented with a paradox. How could you reach the infinite via the finite? If you can't then there should be some kind of gap between the finite and infinite, and yet we never reach such an insurmountable gap. As we continue we simply discover more and more finite numbers. But there must be a largest finite number, right? Again we are thwarted. We find that for every finite number there is yet another larger finite number.

It was the desire to bridge the finite and the infinite that drew me to the subject of large numbers for the first time when I was a kid. I figured if I could achieve the impossible I could resolve the paradoxes that the infinite invoked in me.

We needn't understand the quest to reach large numbers in this way however. Although it's helpful to think of us traveling to a destination that we will never reach, I have later come to realize that we could just as easily think of it as a journey that we can decide when to stop. This avoids the need to invoke the infinite while still allowing us the potential to always continue. But can we always continue? Or must we always stop at some point. If we put the theoretical aside we would have to say that, yes, at some point we are going to have to stop. But if we must stop at some point, and yet we can always seem to go just a little further, just how far can we go then?

It is this question that will motivate the rest of this book: What is the largest counting number we can define?

Some off handed responses are often "infinity", but that can't be because infinity isn't a counting number. Another similar response would be, there is no largest, because for every number we can simply define the next number. This however also can't be true because we have to stop at some point, because we only live a finite amount of time. So then how do we define the largest number that we can define? Certainly it must be larger than any number we can easily come up with like (A), (AA), or (AAA), because we can easily extend these further. By necessity the largest number we could define would require us to use every available resource to us, and it must require and incredible amount of time and work. The problem is we already have failed to produce this number because we have not devoted every resource to it. Even this number, it would seem, is unreachable.

And so this is our quest. We want to define larger and larger counting numbers and see just how far we can get. This can be the only meaningful representation of what we mean by largest defined counting number. As you will see we can get much further than anyone might have expected and the numbers we can reach are so mind-numbingly huge that we can't even attempt to understand their vastness. The frivolous theorem of arithmetic states that "most counting numbers are very very very large". It is the "small" numbers that are rare. This seemingly counter-intuitive truth can be verified by simply considering (C) to be the first "very very very large number". We assume any number larger than (C) is also very very very large. Note that there will only be a finite number of counting numbers smaller than (C), yet there will be an infinite number of counting numbers greater than (C). Therefore there are much more numbers greater than (C) than not and we can say most counting numbers are very very very large.

In the next article we will begin our journey towards the infinite (or from the small and finite if you prefer). This will be the task of the majority of this web-book. We will of course need to discuss other concepts from time to time to build our numbers. At the end of the book I will return to the subject of the finite and infinite and what makes them so incompatible.

Click the links below to return to home, the chapter page, or to the next article...

Next Article: 1.1.4 Jacob's Ladder

[1] http://www.geniebusters.org/915/04e_ex01C.html : This page discusses whether or not Aristotle is in fact the first to "discover" the law of identity.