Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

November 26, 2009

What’s So Special About the Rationals?

Posted by David Corfield

I posed a question over at Math Overflow which netted me one useful answer from Andrew Stacey (along with my first nice question badge). Let’s see if the Café can help me with what I really want to know.

So, the rationals form the unique dense linearly ordered set without endpoints of countably infinite cardinality. On top of this we can build up compatible structure all the way to an ordered topological field. The rationals form the field of fractions of the integers, and the prime field for characteristic 00.

For some additional steps on the way, Wikipedia tells us

By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric d(x,y)=|xy|d(x, y) = |x - y|, and this yields a third topology on \mathbb{Q}. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of \mathbb{Q}. In addition to the absolute value metric mentioned above, there are other metrics which turn \mathbb{Q} into a topological field.

So the question now is whether we can see how the bits of pieces of the structure of the rationals fit together? What are the dependences?

For a simple case of what I’m looking for, take the integers with addition ,+\langle \mathbb{Z}, + \rangle. As the free abelian group on one generator, a ring structure comes for the ride, since we can show that it is a monoid object in the category of abelian groups, using

Hom Ab(F{*}),F{*})Hom Set({*}),UF{*}). Hom_{Ab}(F\{*\}), F\{*\}) \cong Hom_{Set}(\{*\}), U F\{*\}).

So what about the rationals? From the fact that ,<\langle \mathbb{Q}, \lt \rangle is the Fraïssé limit of the category of finite linearly ordered sets and order preserving injections we get density and homogeneity. Could we see how to get a ,+\langle \mathbb{Q}, + \rangle-torsor structure to emerge here?

One thought as to what next: I guess we might consider the Fraïssé limit of finite pointed linearly ordered sets, which should be, I think, ,0,<\langle \mathbb{Q}, 0, \lt \rangle.

Posted at November 26, 2009 5:24 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2118

23 Comments & 0 Trackbacks

Re: What’s So Special About the Rationals?

The fourth or fifth thought that pops into my head is that “since” there’s exactly one two-ways of injectively monotonically mapping [n][n] to [n+1][n+1] such that the two ways are nowhere equal, there should also be (essentially) one way of mapping ,<\langle \mathbb{Q}, &lt; \rangle to itself with no stable bounded intervals.

Hmm… I can get my head around this, though: by trichotomy etc., for any rational xx, ],x[,>\langle ]-,x[, &gt; \rangle and ]x,[,<\langle ]x,-[,&lt;\rangle are both countable infinite d.l.o.w/o.e.p.; so they’re isomorphic; choose such an isomorphism ϕ\phi. Then ϕϕ 1{(x,x)}\phi \cup \phi^{-1}\cup \{(x,x)\} is an order isomorphism of ,<\langle \mathbb{Q},&lt;\rangle and ,>\langle \mathbb{Q},&gt;\rangle, fixing xx. Similarly, there’s (far more than) one such — say ψ\psi — fixing yy for any y>xy&gt;x. Then ψϕ\psi\phi is an order isomorphism mapping x<yx&lt;y to some z>yz&gt;y (and now you know how I write compositions); similarly ϕ 1ψ 1\phi^{-1}\psi^{-1} maps yy to some w<xw&lt; x, and hence ϕ 1ψ 1(x)<x\phi^{-1}\psi^{-1}(x)&lt;x. The union UU of the intervals ](ψϕ) n(x),(ψϕ) n(x)[](\psi\phi)^{-n}(x),(\psi\phi)^n(x)[ is again countable infinite dense linear without endpoints, and stable under ψϕ\psi\phi, which acts as an order automorphism and with no stable (relatively) bounded intervals.

The argument currently playing in my head says I should look at the Euclidean algorithm to generate — from 11, +1{\cdot}+1, and () 1(\cdot)^{-1} — all positive rationals, and then take their negatives … and then argue from some nonsense that ad 1\mathrm{ad}_1 as described, and any order antimorphism inv\mathrm{inv} of U +=U]0,[U^+= U\cap ]0,-[ fixing ad 1(0)\mathrm{ad}_1(0) together generate a useful monoid such that the orbit of 1=ad 1(0)1=\mathrm{ad}_1(0) is a dense order and has no endpoints. It’s then this orbit that becomes the (positive) rationals; but again it’ll be isomorphic to (half of) whatever d.l.o.w/o.e.p. you started with, by structural uniqueness.

Posted by: some guy on the street on November 27, 2009 6:00 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Thanks for this. It took me a moment or two to work out that d.l.o.w/o.e.p. = dense linear order without end points.

Is this right? You’re tapping into the idea of two reflections generating a translation in the rigid case. Your xx is going to be the 00, and your yy the 1/21/2. The choice of an order inversion around xx then one about yy gives a shift to the right, which you call addition by 11.

Posted by: David Corfield on November 27, 2009 10:51 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

I think jumping on y=12y=\frac{1}{2} is a bit premature; but yes, basically I’m trying to leverage something from the infinite dihedral group. The wrinkle is that D D_\infty can act on a dense linear order in many ways; in particular, there’s no reason to suppose the subset where it doesn’t act freely is discrete in the order topology; so that’s the reason for taking the interval closure of one orbit of the subgroup of translations.

Reflecting on it in daylight, this boils down to a big lemma saying that a dense linear order has an infinite discrete subset and this more-than-free action of \mathbb{Z} (could we call it Archimedean?). It occurs to me now that since ×\mathbb{Q}\simeq\mathbb{Q}\times\mathbb{Z} (antilex ordering) we don’t really need to prove it that way; but I was rather hoping to minimize the order-type of all the choices needed.

Posted by: some guy on the street on November 27, 2009 5:31 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

I wonder how far one could proceed by considering the symmetries of the feltered subcategory of finite linear orders and embeddings, or perhaps subcategories of that.

Posted by: David Corfield on November 27, 2009 11:12 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Are you trying to get the abelian group and ring structure on \mathbb{Q} to arise from its structure as a Fraissé limit? This remark of yours is obscure to me:

Could we see how to get a ⟨\mathbb{Q},+⟩-torsor structure to emerge here?

If you’re trying to get a ⟨\mathbb{Q},+⟩ torsor structure on something, you must already know \mathbb{Q} is an additive group. But then it’s automatically torsor of itself, so trying to use Fraisse limit considerations to make some isomorphic set into a ⟨\mathbb{Q},+⟩ torsor seems like an unnecessary acrobatic feat.

Of course, mathematics often profits from unnecessary acrobatic feats. But maybe I just don’t understand your starting-point and goal.

Just naively, it strikes me as strange to focus so much attention on the order structure of the rationals if you’re really trying to get to the bottom of the question “What’s so special about the rationals?” I guess it can lead to some interesting questions… but to me, and probably most mathematicians, what’s always been so special about the rationals is that they’re field of fractions of the integers.

Somehow your focus on the order structure of the rationals reminds me of this puzzle, which you may enjoy:

The rationals are a dense linearly ordered set with 0\aleph_0 points between any two distinct points. It’s an ordered field.

The real numbers are a dense linearly ordered set with 2 02^{\aleph_0} points between any two distinct points. It’s an ordered field.

Take your favorite cardinal. Is there a dense linearly ordered set with κ\kappa points between any two distinct points? Is it unique, as a linearly ordered set? And: can you make it into an ordered field?

In other words, does the family ,,...\mathbb{Q}, \mathbb{R}, ... have other members?

Posted by: John Baez on November 27, 2009 6:44 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Take your favorite cardinal.

OK, taken.

Is there a dense linearly ordered set with κ points between any two distinct points?

Yes.

Is it unique, as a linearly ordered set?

Yes.

And: can you make it into an ordered field?

No, I'd have to pick an infinite cardinal to do that!

Posted by: Toby Bartels on November 27, 2009 7:30 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Toby wrote:

No, I’d have to pick an infinite cardinal to do that!

So your need for the cardinal to be infinite kicked in just now? What about a dense linearly ordered set with 7 points between any two distinct points? Either we mean ‘exactly 7 points’, in which case it’s unique but doesn’t exist, or ‘at least 7 points’, in which case it exists but isn’t unique. No?

But enough of this goofing around. Of course I meant an infinite cardinal κ\kappa. And I meant exactly κ\kappa points between any two distinct points.

What if κ=2 2 0\kappa = 2^{2^{\aleph_0}}? Can we get an ordered field in this case?

Posted by: John Baez on November 27, 2009 7:47 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

The only non-infinite cardinal κ\kappa that I can think of for which there is a dense linearly ordered set with κ\kappa points between any two distinct points is zero.

Posted by: Mike Shulman on November 27, 2009 8:28 PM | Permalink | PGP Sig | Reply to this

Re: What’s So Special About the Rationals?

Of course Toby should have thought of that example, and knowing him, I thought of it too.

And in this case we get the field with one element.

Posted by: John Baez on November 27, 2009 8:35 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Of course I meant an infinite cardinal κ\kappa.

You may have meant that, but you said ‘Take your favorite’. And it's the only finite cardinal for which the answer to the first two questions is affirmative!

And in this case we get the field with one element.

I thought of that, but it seemed funny to say that the field with one element has no elements.

But enough of this goofing around.

OK, a more serious partial answer: The (large) field of surreal numbers is a kind of Fraïssé limit of all (small) ordered fields. Can we find what we want inside it?

Posted by: Toby Bartels on November 27, 2009 8:55 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Toby wrote:

I thought of that, but it seemed funny to say that the field with one element has no elements.

Wait a minute. You seem to be hinting that the empty set is a dense linear order with exactly 0 points between any two distinct points. And I guess that’s correct.

But are you saying the one-element set is not a dense linear order with exactly 0 points between any two distinct points? I thought it was. That’s why I cracked a stupid joke about the ‘field with one element’.

Anyway… this the conversation has now sunk to sad depths of silliness. I actually thought David might be interested in pondering the possibility of ordered fields like \mathbb{Q} and \mathbb{R}, but with various other (infinite) numbers of elements between distinct points. He’s being paid to talk about infinity, after all.

Posted by: John Baez on November 27, 2009 9:07 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

So where are we? We know by the Löwenheim–Skolem theorem that if a countable first-order theory has an infinite model, then for every infinite cardinal number it has a model of that size. So there is a densely linearly ordered field for each cardinality.

Do we know about categoricity? Well, we know that

The theory DLO of dense linear orders with no endpoints (i.e. no smallest or largest element) is complete, ω\omega-categorical, but not categorical for any uncountable cardinal.

I doubt adding in field structure would help. Already that’s going to distinguish \mathbb{Q} and (2)\mathbb{Q}(\sqrt{2}).

One of the motivations for this thread was a comment by a philosopher to a talk given on category theory that it’s all very well describing general principles, but does it deliver specific structures? This is really what I’d like to know about. Universality seems to be category theory’s most powerful way of getting at specificity. I find it interesting when universality buys you other properties.

E.g., why should SetSet as the free cocomplete category on 11 be a topos?

Posted by: David Corfield on November 29, 2009 7:12 PM | Permalink | Reply to this

Re: Large properties as Set (Was: What’s So Special About the Rationals?)

E.g., why should SetSet as the free cocomplete category on 11 be a topos?

That's a good question, and my answer is: because SetSet is a topos.

This is almost begging the question, but it's not; the question is ‹Why is the free cocomplete category on 11 a topos?›, and the answer is ‹because SetSet is a topos›, which is not the same.

Notice that SetSet appears twice in the question as you phrased it: once as simply an identifier for the free cocomplete category on 11, and once as part of the definition of cocompleteness. And that's why it appears in the answer.

An analogous answer also explains why SetSet has a natural numbers object, why SetSet is cartesian closed, etc. In each case, one can find unconventional mathematicians (constructivists, finitists, predicativists) who don't believe this property of SetSet and therefore don't believe it of the free cocomplete category on 11; and you can prove that the free cocomplete category on 11 has such a property directly (that is, not by first simply proving that this category is equivalent to SetSet) by applying the cocompleteness condition.

Posted by: Toby Bartels on November 29, 2009 8:27 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

But are you saying the one-element set is not a dense linear order with exactly 0 points between any two distinct points?

Ah, so it is! I was imagining a missing clause that the entire field had to have cardinality κ. That's not there, so the set in question is not unique!

Anyway… this the conversation has now sunk to sad depths of silliness.

No, I had something serious to say, about surreal numbers. You may have missed it in the silliness. Unfortunately, it wasn't very deep.

Posted by: Toby Bartels on November 28, 2009 1:01 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

I saw your point about the surreal numbers, and that’s closer to what I’d actually like to talk about!

However, since you were engaged in a bout of overly logical thinking, I could not resist checking whether your logic circuits had a small flaw.

Posted by: John Baez on November 28, 2009 1:31 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

There is a way to get a torsor without first having a group, namely by having a heap. Any heap gives rise to an “automorphism group” over which it is then automatically a torsor. I’m not sure if that’s what David had in mind, though.

Posted by: Mike Shulman on November 27, 2009 8:30 PM | Permalink | PGP Sig | Reply to this

Re: What’s So Special About the Rationals?

Of course, mathematics often profits from unnecessary acrobatic feats. But maybe I just don’t understand your starting-point and goal.

I’m playing with the idea that certain structures act like ‘attractors’ in that although simply defined, they possess a long list of other properties. This is a phenomenon Hazewinkel observed in his Niceness Theorems:

It appears that many important mathematical objects (including counterexamples) are unreasonably nice, beautiful and elegant. They tend to have (many) more (nice) properties and extra bits of structure than one would a priori expect.

Hazewinkel’s star example is the ring of symmetric functions in an infinity of indeterminates, which is extraordinarily ‘nice’. He tries to relate these different properties to each other.

So here I was trying to do something similar for the rationals. Knowing the rationals as the field of functions of the integers, do we know that they are linearly ordered, and as an order, the Fraisse limit of finite orders?

Well, yes to the first part. The field of fractions of an ordered ring is ordered. Maybe the second is surprising as there could be no ring structure compatible with the order on the finite ordered sets, so the capacity to receive a ring structure ‘emerges’. I was wondering if the emergent homogeneity of the Fraïssé limit had something to do with it. A topological group’s going to have to have that.

Posted by: David Corfield on November 28, 2009 3:32 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Speaking of Fraissé limits and the rationals, I just bumped into a curious paper entitled Towards Combining Dense Linear Order with Random Graph. I don’t understand it.

Posted by: John Baez on November 27, 2009 8:52 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

If you want to discuss it, I can try to be your sparring partner until a real expert joins us.
What is the first statement that puzzles you?
(Educated guess: A mathematician would probably misunderstand “module” in the introduction. I’m pretty sure that they don’t mean the algebraic object, but a “software module”).

Posted by: Tim vB on November 30, 2009 8:33 AM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

The significance of Q is that it’s the injective hull of Z. The fact that it’s a field is an incidental consequence of this more fundamental significance.

Posted by: Walt on November 29, 2009 10:18 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

I just noticed injective hulls in Section 9 of The Joy of Cats.

Posted by: Toby Bartels on November 29, 2009 11:26 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

Earlier it has been shown that many familiar constructions, particularly “completions” such as the completion of a metric space or the Cech-Stone compactification of a Tychonoff space, can be naturally regarded as reflections. However, there also exist familiar completions that cannot be (or only artificially can be) regarded as such. Examples are the Mac Neille completion of a poset and the algebraic closure of a field. In both cases, and in several others, the construction in question can be regarded rather naturally as an injective hull, a concept that will be studied in this section. (p. 152)

Sounds like this chapter might also help with the organisation of completion.

Posted by: David Corfield on November 30, 2009 2:01 PM | Permalink | Reply to this

Re: What’s So Special About the Rationals?

I started an nLab injective hull page, which needs some help. It seems to cover a range of ‘completions’.

Posted by: David Corfield on December 1, 2009 8:08 AM | Permalink | Reply to this

Post a New Comment