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Ordinals, Cardinals, etc. (Part 1 – Foundations)

by on November 18, 2011

In my reading of Hovey’s “Model Categories,” Hovey talks a bit about ordinals, cardinals, transfinite compositions, small objects and ordinals being \gamma filtered.  I found that I didn’t have a very good intuition for such situations, and thought it might be helpful to write a short article on it, since such issues, while often ignored or downplayed, become relevant in the category theoretic facet of homotopy theory. One note: some of this early stuff I’m doing from memory, so excuse any notational discrepancies, I think the basic idea is still there. And also, we will, without even mentioning it, be assuming the axiom of choice here.

(Editor’s Note: This first part just contains foundational stuff on ordinals and cardinals, i.e. what are they?  The next piece of this article will actually get into Hovey’s stuff.)

First of all, I’d like to briefly discuss the construction of the ordinals.  We can consider an ordinal “number” \alpha to be the set of all “smaller” ordinals.  Of course, without any background, the above statement doesn’t make any sense.  So, let’s start at the beginning, i.e. the empty set \emptyset.  This is our first ordinal.  Next we take the set containing the empty set, which we might write as \emptyset+1 or, if we’re feeling adventurous, just 1. The next ordinal will be the set containing all smaller ordinals, i.e. \{\emptyset, 1\} or maybe just 2.  We can continue this process ad infinitum.  One thing to note here is that when we look at the cardinality of these ordinals (i.e. how many elements are in each, since each is a set) we’re not counting the cardinality of the sets the ordinal contains.  We’re counting each contained set as one object in itself.  Thus while the next ordinal could be written as \{\emptyset, 1, 2\} it would be 3 since it contains three elements.

Now, unless we’re just doing our taxes, we’d like a way to enumerate things that aren’t finite (i.e. infinite…).  To do this we take a “limit ordinal,” (up till now we’ve been taking what are called “successor ordinals”).  We let \omega just be the set of all the finite ordinals, which we note is in bijection with \mathbb{N},\mathbb{Z} and \mathbb{Q}. This ordinal is, as you probably know, the first of the countable (and infinite) ordinals.  Of course, continuing the process from the finite ordinals, we might now take the set of all finite ordinals AND \omega, which we denote by \omega+1.  Note that \omega+1 is a successor ordinal again, since we got it by just bumping up a step, instead of taking the union of some infinite sequence of things (some might say “But we are taking the union of an infinite number of things, because \omega+1 has both \omega in it and all the finite ordinals,” but the idea is that we’ve already dealt with that process by jumping up to \omega and can just start taking successor ordinals again). Similarly to what we did before, we can now just keep taking successor ordinals, getting \omega+k for every k\in\mathbb{N}.  However, this is not enough for us now.  So again we take a limit ordinal, this time taking the union of all the finite ordinals, \omega, \omega+1…. and call this thing \omega+\omega or 2\omega.  Of course we can now repeat, getting \omega\cdot\omega, \omega^\omega and so on and so forth.

The thing to note here is that all of the things we’ve gotten above, and infinity of them in addition to the ones we’ve talked about, have one thing in common.  As sets, they are all in bijection with \omega.  That is, they’re all countable (there is lots of interesting stuff going on with ordinals, and lots of interesting definitions; check out wikipedia to follow that particular path, specifically Cantor-Normal form, Church Kleene Ordinal and recursive ordinals are rather interesting).  So, for any one of those ordinals, say \alpha, we know that \vert\alpha\vert=\aleph_0 (except for finite ordinals, which have cardinality themselves, and so are in fact also the finite cardinals), which is by definition, the cardinality of something which is “countable.” In general, the cardinality of an ordinal \alpha is defined to be the smallest ordinal which is in bijection with \alpha.  Hence \aleph_0=\omega from our point of view.

So, like we’ve done before, let’s take the union of all the countable ordinals! We do, and what we get is what we denote as the set \omega_1.  This is the smallest uncountable ordinal, and so we also call it \aleph_1.  The point is, this process just keeps going, forever. I think that what we’ve done so far however will be enough to serve intuition.

As a side note, the content of the Continuum Hypothesis, which is something I struggled with for a time, is that 2^{\aleph_0}=\aleph_1.  That is, the cardinality of the power set of \omega is in fact \omega_1.  It is necessarily true that the cardinality of \aleph_1 is less than or equal to 2^{\aleph_0} since in some sense \aleph_1 is the next biggest cardinal after \aleph_0 and that’s the way we constructed it, but showing that they are equal is independent of ZFC.




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