We can get at another infinite cardinal that is greater than by thinking about the ordinal numbers. Think of the natural, or counting numbers . These numbers double as the finite cardinal numbers. Cardinal numbers, we have seen, express the size of a set or the number of objects in a collection (e.g., as in “24 is the number of hours in a day”). But they also double as the finite ordinal numbers, which indicate a place in an ordering or in a sequence (e.g., as in ” the letter ‘x’ is the 24th letter in the English alphabet”). In the case of finite collections, the finite ordinal numbers are the same as the finite cardinals.
But when we start thinking of infinite collections the similarities diverge. In order to see the differences in the infinite case we should get clear on what an ordinal number is. Say that a set is transitive if, and only if, every element of is a subset of , where is not a urelement (something that is not a set). Now say that a set is well-ordered by the membership relation, , if . What this does is simply order the elements of in terms of membership. We can do this type of thing with transitive sets. Combining these two definitions we get the definition of an ordinal number: a set is an ordinal if, and only if, is transitive and well-ordered by .
Now, let (i.e., the set of natural numbers). is an ordinal because when we think of the natural numbers as constructed by letting and letting , the singleton set of , , and so on and so forth it satisfies the definition above of an ordinal number as a transitive set well-ordered by . So we can set up the sequence with all ordinals less than either equal to or one of its successors.
Suppose that you take the natural numbers and re-arrange (re-order) them so that is the last element. This is weird because the regular ordering of the natural numbers has no last element. But still, you can think of there being a countable infinity of natural numbers prior to the appearance of . So we have the standard order of and we have added another element, . If we let be the standard order of , then we have just described . We can do the same thing by now setting as the last element of and thus get . Note that addition (and multiplication below) does not commute; e.g., and not . This process can be generalized (e.g., ) to get .
Again, doing something weird: take the natural numbers and put all the even numbers first, followed by the odd numbers. It’ll look like this, . Here we have taken (the evens) and appended it to (the odds) so we have . In each case, , and , we are dealing with the same cardinality, the cardinality of .
We’ve created a variety of different ordinal numbers here, and, as they represent different orderings of the natural numbers, they are all countable. There are many more countably infinite ordinals. For example, the ordinal .
Taking the countable ordinals and laying them out (kind of like in the previous sentence but starting with ) we end up with a a set that is itself an ordinal. In order to see this let be the set of countable ordinals. If then since the members of are countable ordinals. Therefore is an ordinal.
It is in fact the first uncountable ordinal because if it were countable, then would be a member of itself and there would be an infinitely descending sequence of ordinals. But because the ordinals are transitive sets (see definition above), this cannot be the case. So the set of countable ordinals is uncountable. (It is also the smallest such set because the ordinals are well ordered by , so every ordinal in is a member of and countable.) This uncountable set goes by the name .
Here we see how the similarities between the ordinal numbers and the cardinal numbers in the finite case diverge in the infinite case. Whereas there is only one countably infinite cardinal, , there are uncountably many countably infinite ordinals, namely all countably infinite ordinals less than .
It is natural to wonder about the cardinality of the set of countable ordinals. Its cardinality is transfinite and is denoted by the uncountable cardinal number, .
So far we’ve talked about (see here, and here, respectively), and have generated by considering the uncountable set of countably infinite ordinals, . In the next update we’ll talk more about the relationship between these cardinal numbers as well as the celebrated axiom of choice.