# Cardinal vs. ordinal: When learned? When needed?

Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between "one" and "first") in pre-college or early-college math instruction outside of set theory? If so, where is it used? Where is it needed?

I am asking because I don't see a need for the distinction until one encounters infinite sets, say, ordering the algebraic numbers. In my own college teaching I need to distinguish between countably infinite $$\aleph_0$$ and uncountably infinite $$\mathfrak{c}$$, but rarely do I use $$\omega$$. It would help if I understood where the cardinal vs. ordinal distinction is used/needed.

• I teach the distinction in my courses for future elementary teachers. Someone else has a discussion here, similar to what I discuss: christopherdanielson.wordpress.com/2010/09/15/… I like to stress that the ordered list of natural numbers (1,2,3, 4, ...) is useful because we can put initial segments of it in one to one correspondence with sets of objects (this is "counting"). – Steven Gubkin Dec 26 '19 at 20:08

Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between "one" and "first") in pre-college or early-college math instruction outside of set theory? If so, where is it used? Where is it needed?

I'm not aware of any early-college courses in which the distinction (or either notion individually) is taught systematically. I've seen it come up in "introduction to proof" courses, and I could imagine it appearing in a discrete math course, but syllabi for those courses tend to vary a lot between institutions and instructors.

Any sufficiently advanced logic course, whether or not it includes set theory, is going to have to cover both notions, but that's getting out of what I'd call "early-college".

In my own college teaching I need to distinguish between countably infinite ℵ0 and uncountably infinite 𝔠, but rarely do I use 𝜔. It would help if I understood where the cardinal vs. ordinal distinction is used/needed.

I have trouble imagining where the distinction itself is useful. The question is where one wants to work with ordinals at all; I think the main reason people usually introduce ordinals is because they need to do transfinite induction.

• Good point re transfinite induction. Thanks. – Joseph O'Rourke Dec 19 '19 at 11:43
• FWIW, I just looked at the two discrete math books I have here (I believe the two most popular), and the cardinal/ordinal terms do not appear in either one. (Except for one saying "cardinality comes from the common usage of the term cardinal number".) – Daniel R. Collins Dec 22 '19 at 7:07
• "I think the main reason people usually introduce ordinals is because they need to do transfinite induction." Additional note: this doesn't require one to name any specific ordinals either, so even here there's no need to actually discuss $\omega$ or any specific ordinal, unless there are edge cases you may need to cover. – Simply Beautiful Art Jan 5 '20 at 23:27

I don't know if this is always in a set theory course or not, but it makes a difference when you define the natural numbers using the Peano axioms (which is ordinal) or the Frege construction (which is cardinal).

Here in the US, the words began to be commonly taught in the early grades during the "New Math" era. I remember in first and second grade (ca. 1972) being taught this distinction and thinking that it was dumb. In this context, kids were taught that "cardinal" numbers were for counting how many things you had, while "ordinal" numbers were numbers like "first," "second," etc., which told you an item's position in a list. This may have mostly died out along with other goofy and ill-considered features of New Math, but if you google, you can still find materials online such as worksheets where kids are taught the two terms and asked to match up 3 with "third," etc.

And of course if some piece of silliness like this makes it into the lower-grade curriculum, it becomes something that has to be taught to preservice elemetary teachers as part of their education coursework. They don't have any idea of the context where this distinction would actually be useful, with infinite sets and so on. They just parrot it like all the other crap they are forced to parrot in the classes taught by a university's education department. So if your question is when this distinction is learned and when it's needed, one answer is that it may be learned when people are studying to get a teaching credential, and needed when they need to take a multiple-choice test that proves they know enough math to be elementary school teachers.

There may be a more general phenomenon here, which is the perceived need to make isomorphisms more explicit, even when nothing is gained thereby. For example, Euclid didn't need separate concepts of angle and measure-of-an-angle, but it seems to be standard these days in high school geometry books to treat these as separate concepts, and to introduce a postulate saying that there is a mapping from angles to real numbers, which has certain properties.