What do we mean by Numbers? A simple introduction to Set Theory
The vast majority of people never think about what they mean when they use numbers to describe things. The concept is so ingrained into our early development that we simply take for granted the fact that people understand us when we apply numbers to different things. For most people, numbers are simply numbers, and questions about the meaning of those numbers are confusing and seemingly nonsensical.
Mathematicians are not most people.
For quite a long time, now, mathematicians have recognized that there are at least two very distinct ways in which we use numbers to describe things. Being the scholarly, academic types that they are, mathematicians have assigned names to these two different types of numbers which sound heady and difficult to the average person: ordinal numbers and cardinal numbers. Indeed, even mathematics students sometimes need quite a bit of work and explanation in order to really grasp the difference between these two types of number; but I’m going to do my best to explain these things in a very simple way for a casual audience.
Ordinal and cardinal numbers roughly correspond to the ideas of value and size, respectively.
Everyone can fairly intuitively recognize that value and size are entirely distinct concepts. If I have a bar of lead and a bar of gold which are both 30 cubic centimeters, it doesn’t take much to realize that both are the same size. However, if I were to allow you to choose one to take, who would hesitate to grab the gold bar? Despite being equal in size, the two bars are certainly not equal in value. The gold has far, far greater value than does the lead. What most people do not realize is that there is a similar concept to be found among numbers.
In order to explain, I’m going to ask you to take a look at the picture below.
In this picture, we have a set of ravens and a set of owls. Without counting or using numbers to describe either set, how can we know that both sets are the same size? Most people will immediately intuit that this is the case, even before recognizing that there are 5 of each, ravens and owls. However, those same people usually don’t understand why. You see, our brains tend to naturally make a connection that we might not consciously recognize at first. We notice, without even realizing it, that for each raven there is also an owl. We can match them up in what mathematicians refer to as a one-to-one relationship, as in the image below.
When the elements of two sets can be placed into a one-to-one relationship, like this, we say that they have the same cardinality. If they cannot, then the cardinality of the two sets is different. As such, cardinality is a means of discussing the size of a set.
However, as mentioned, there is another way in which we think about numbers, too. Have a look at the following image. In it, you’ll see a series of symbols being compared to one another using less-than symbols (<). Again, without assigning any numbers to the symbols, can you place them in order from least to greatest?
It’s not a very difficult puzzle. We see that the Mirror is less than the Mask, but the Wolf is less than the Mirror. Similarly, the Rose is less than the Scepter, but the Mask is less than the Rose. All it takes is a little rearrangement of the information which we have been given to see the following:
Without using any numbers, at all, we’ve placed these unfamiliar symbols into order from least to greatest. We don’t need any numbers to know, for example, that Wolf is less than Mask or that Mirror is less than Scepter. In short, we’ve established an order of value for these symbols, without needing to know anything else about them, at all. This sense of ordering things by their value is what we mean when we discuss ordinal numbers.
Thus far, everything we’ve discussed should seem fairly simple and straightforward. You might, in fact, wonder why I’ve even chosen to discuss it, at all. Sure, ordinality and cardinality might be two different concepts, but what difference does it really make? One does not need to consciously understand these differences in order to talk about numbers in a meaningful way.
Indeed, when we are discussing the numbers with which most people are quite familiar, saying that ordinal numbers are different than cardinal numbers seems like a distinction without a real difference. Who cares if we are talking about the cardinal number 5 or the ordinal number 5, for instance? Isn’t the number 5 just the number 5 regardless of whether we’re talking about ordinality or cardinality?
In the case of finite sets, this holds quite well. The cardinal 5 and the ordinal 5 aren’t really very different, other than in some minor quality about how we use them. The whole notion seems like quibbling pedantry, to the layperson. The problem arises, however, when we begin talking about infinite sets. It does not take very long to realize that with infinite sets, ordinals and cardinals are very different things.
Let me explain by discussing one of the ways in which we can construct the Natural numbers. We begin with an empty set– that is, a set which contains no elements. Now, we can play a little game with one very simple rule. The rule is that we are allowed to create a new set called a Successor which contains as its elements every other set which we already have. Since we currently only have the empty set, this means that we can create a new Successor set which contains the empty set as its only element. Once we have done that, we can apply the rule again, this time creating a Successor set which contains the empty set and the set containing the empty set. We can then apply the rule again to find another Successor, and we can continue this process ad infinitum.
If this seems confusing, take a look at the image below. The boxes represent our sets, while commas separate the elements within a set.
Now, all this “Successor to the Successor to the…” language is the confusing part of all this, so let’s make things easier by assigning symbols to represent each of the sets which we have created. We can assign the symbol “0” to the Empty Set, the symbol “1” to the Successor to 0, the symbol “2” to the Successor to 1, and the symbol “3” for the Successor to 2. I’m sure you now see where I am going with this.
If we continue playing our Successor game, we can probably already see that the set “4” will contain 0, 1, 2, and 3. The set “5” will have sets 0, 1, 2, 3, and 4 as its elements. The set “6” will be constructed of elements 0, 1, 2, 3, 4, and 5. Et cetera, et cetera. This is all very straightforward. All of the numbers which are created in this way are termed the Natural numbers by mathematicians.
This setup gives us a clear method for discussing the ordinality, or value (as I have been describing it), of a set. A set is said to be “less than” another set if the former is an element of the latter. So, 1 is less than 3 because 3 contains 1 as an element, in this construction. Similarly, 0 is less than all of the other sets because 0 is an element of all of the other sets. Conversely, 2 is not less than 1 because 2 is not an element of 1; and no Natural number is less than 0 because no Natural number is an element of 0.
We can also use this method to define what it means for sets to be ordinally equal. If we have two sets, A and B, we can say that A and B are ordinally equal if A is not less than B and B is not less than A. So, for example, if we were to compare our set “3” to a copy of itself, we would see that original-3 is not less than copy-3 (because the set “3” does not contain 3 as an element), and copy-3 is not less than original-3 for the same reason. Therefore, we would conclude that 3 is ordinally equal to 3.
We can also use this setup as a very easy way of discussing the cardinality, or size, of a set. For an illustration of this, take a look at the next image.
Let’s say that we want to compare the set which we called “3” with the set of owls, pictured here. If you will recall from our discussion of ravens and owls earlier, if the elements of two sets can be placed in a one-to-one relationship with one another, then those two sets have the same cardinality. So our set of owls, in this picture, has the same cardinality as our set “3” has because we can match each of the owls to each of the elements in 3. When two sets have the same cardinality we can say that they are cardinally equal.
With the Natural numbers, it is plain to see that two sets which are ordinally equal are always going to be cardinally equal, as well. The set 3, for example, is not less than itself and it can have its elements matched one-to-one with itself. Similarly, if two numbers are not ordinally equal they will not be cardinally equal either. So, 3 is less than 4, and there is no way to match up all of the elements of 3 and 4 so that they will be in a one-to-one relationship.
This all may seem rather obvious to some. Of course two sets which are ordinally equal are going to be cardinally equal! How could things be any other way? One might wonder anyone would bother to worry about the distinction between ordinality and cardinality, at all. The truth is, however, that numbers are far stranger than most people will ever realize.
Let’s say that I create a new set which contains as its elements all of the Natural numbers. We will call this set “ω” (the Greek letter omega). By the definition of “less than” which we used earlier, it is clear that every Natural number is less than ω, since every Natural number is an element of ω. Similarly, we see that ω is not less than any Natural number, since no Natural number would have ω as an element. We have created a completely new ordinal number.
Now, we can apply our Successor rule again to create a set which contains all of the Natural numbers and ω as its elements. We’ll call this Successor set “ω+1.” Clearly, since ω is an element of ω+1, we can see that ω is less than ω+1. Again, we can continue to apply the Successor rule to create new sets which we might call “ω+2,” “ω+3,” “ω+4,” and so on, and we can continue to see that our ordinal “less than” rule still holds for each of these. We call ω and the Successors which can be generated from it transfinite ordinals.
One might easily think that, just as was the case with the Natural numbers, if two transfinite sets are not ordinally equal, then they can’t be cardinally equal, either. But, as we shall see, the world of the infinite is a very, very different place.
Now, it’s clear that ω does not have the same cardinality as any of the Natural numbers, because no matter which Natural number one chooses, attempting to set up a one-to-one relationship between its elements and those of ω will result in unmatched elements remaining in ω; and given all the precedent which we saw in the Natural numbers, we might expect that ω+1 has a different cardinality than does ω. Indeed, at first glance this might seem to be the case. When we try matching the elements of ω to those of ω+1, the obvious thing would be to match 0 to 0, 1 to 1, 2 to 2, and so on. If we were to go about it in this way, we would match up every Natural number in the set ω to its counterpart in ω+1 and there would still be one unmatched element in ω+1.
However, what if we were to be a little bit clever? Instead of matching each Natural number to its counterpart in the other set, let’s instead match the ω-element of ω+1 with the 0-element of ω. We can then match each Natural number element in ω+1 with its Successor element in ω, so that 0 matches with 1, 1 matches with 2, et cetera. Now, we find every element in ω+1 does pair up with an element in ω in a one-to-one relationship.
Despite the fact that ω is ordinally less than ω+1, we can see that these two sets are cardinally equal! In fact, ω has the same cardinality as any of the Successor sets which can be generated from it. So, ω has the same cardinality as ω+1 or ω+15 or ω+3672. Mathematicians refer to the cardinality of ω with another symbol, (pronounced “Aleph null” or “Aleph zero,” from the Hebrew letter). So, despite the fact that the ordinal number 3 and the cardinal number 3 don’t seem to be very different, it becomes exceedingly clear that the ordinal number ω and the cardinal number are not the same thing, at all.
A bar of lead and a bar of gold might be the same size, but the bar of lead has far less value than the bar of gold. Similarly, we now see that it is entirely possible for two sets to be the same size, even if one has far less value than the other. Though we do not usually see it in our everyday experience, there is a very distinct and meaningful difference between numbers which are used ordinally and numbers which are used cardinally. Despite the fact that the concept is so thoroughly ingrained in us, it is not always obvious what it is that we mean when we use numbers.