# I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ? Note : I am an absolute novice, and I'm having a little trouble visualizing ordinal numbers.

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Where did you read that (esp. ordinals=length)? Better: Ordinals refer not only to the size of a set, but also how its elements are ordered. – Hagen von Eitzen Jul 21 '13 at 11:37
This is the link:ls.poly.edu/~jbain/Cat/lectures/07.OrdsandCards.pdf. – Novice Jul 21 '13 at 11:38

If the line to the bathroom is finite, then the length and the number of people in that line is the same number.

However suppose that the line is infinite, but every person has to wait a finite time before entering the bathroom. Now comes another guy, and he has to wait until all those before him get in before he can use the facilities.

In terms of cardinality we didn't increase the line, but its length is longer by one person now.

Formally, the infinite line where every person has to wait a finite amount of time is $\Bbb N$ with the order $\leq$. Indeed every natural number has only finitely many predecessors. The next line is $\Bbb N\cup\{\bullet\}$, where $\bullet$ is some new element which is not a natural number and we decree that it is larger than all the natural numbers themselves.

So the order type of the new set is longer because it has an initial segment of order type $\Bbb N$, but another point above that. However the cardinality is the same. To see that note that the following map is a bijection between the two sets: $$f(n)=\begin{cases}\bullet & n=0\\ n-1 & n\neq 0\end{cases}$$

So we have a strictly longer line (and this poor guy who has to wait an infinite time before he can pee), but that line has the same number of people.

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That was really illuminating, finally understood the difference between ordinality and cardinality. Thanks a lot :) – Novice Jul 21 '13 at 11:44
@Novice: You're welcome. – Asaf Karagila Jul 21 '13 at 12:20
The sanitary situation would be even worse if people were to enter largest number first. :) – Hagen von Eitzen Jul 21 '13 at 12:26

Example $6$ at the bottom of the first page of that PDF is a good example of the difference between size, on the one hand, and length and shape on the other, in the sense in which its author means those terms. Consider the following two ways of arranging the natural numbers. First, in their usual order:

$$\begin{array}{cc} 0&1&2&3&4&5&6&7&8&9&\cdots \end{array}\tag{1}$$

Then with the even numbers first, in their usual order, followed by the odd numbers in their usual order:

$$\begin{array}{cc} 0&2&4&6&8&\cdots&1&3&5&7&9&\cdots\\ \end{array}\tag{2}$$

It’s the same set $\Bbb N$ in both cases, so the in some sense two arrangements are certainly the same size. On the other hand, if we set them side by side, starting at the lefthand end, something very different happens:

$$\begin{array}{cc} 0&1&2&3&4&\cdots&n&\cdots&|\\ \hline\\ 0&2&4&6&8&\cdots&2n&\cdots&|&1&3&5&7&9&\cdots\\ \end{array}\tag{3}$$

The parts to the left of the vertical line match up perfectly, and there’s nothing in the top row to match up with the bottom row. Even though the top and bottom lines are arrangments of exactly the same set $\Bbb N$, in terms of the arrangement the bottom line is somehow twice as long as the top line. And yes, the unmatched part of the bottom line really does have the same shape as the matched part, as you can see from the following arrangement:

$$\begin{array}{cc} 0&1&2&3&4&5&6&7&\cdots&n&\cdots\\ \hline 0&2&4&6&8&10&12&14&\cdots&2n&\cdots\\ \hline 1&3&5&7&9&11&13&15&\cdots&2n+1&\cdots \end{array}\tag{4}$$

In more formal terminology, what the author calls size is known as cardinality. Two sets are said to have the same cardinality if there is a way to pair them up one for one, with nothing left over in either set; in technical terms, this means that there is a bijection between the two sets, a function that is one-to-one and onto. Clearly $\Bbb N$ can be paired up with itself in this fashion no matter how we rearrange it.

Length and shape are informal ways of talking about what is properly called the order type of a linear arrangement of a set. Let $E$ be the set of even natural numbers and $O$ the set of odd natural numbers. $(4)$ shows that we can pair up $\Bbb N$ with each of $E$ and $O$, and $E$ and $O$ with each other, in a one for one fashion, so these three sets all have the same cardinality (‘size’). On the other hand, we can pair up $\Bbb N$ with $E$ and then stick $O$ (in its usual order) after $E$, as in $(3)$ to get an arrangement that looks like two copies of $\Bbb N$ placed end to end:

$$\begin{array}{cc} 0&1&2&3&4&\cdots&n&\cdots&|&0&1&2&3&4&\cdots\\ \hline\\ 0&2&4&6&8&\cdots&2n&\cdots&|&1&3&5&7&9&\cdots\\ \end{array}\tag{5}$$

In terms of order type the arrangement $(2)$ is twice as long as $(1)$: it can be matched up one for one, preserving the order, with two copies of $(1)$ placed end to end, as in $(5)$.

This is all very informal, of course, but that may be helpful in getting started with the ideas.

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