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There are a number of notions of infinity in mathematics that are respectable. One of the first is 'the point at infinity' to the line or plane; but one can argue that this is a spurious infinity as in another perspective this simply closes up the line to make a circle; and the plane to make the surface of a sphere.

In mathematics this construction is called 'compactification'; as the spherical surface is 'compact' in a way that the infinitely extensive plane is not.

Do the ordinal infinities or cardinal infinities allow a similar perspective?

notes

*. It 'looks' possible to embed the long line into the Hilbert cube; which though having a countable number of dimensions is formally compact in another standard topology.

  • There are many compactifications, of which the 1-point compactification is an extremely simple one. See for example. en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification I don't know what you mean by "for real" or "respectible." You seem to have a particular point of view here. Certainly ordinals and cardinals are respectable and for real in modern mathematics, as you well know. Are you wearing your finitist hat today? – user4894 Sep 24 '14 at 0:27
  • Sure, and why should that preclude asking for compactifications of ordinal or cardinal infinities? Compact objects are more respectable because we can do more with them - its suprising just how often after an infinite object is constructed it is compactified or 'cut down' to make it possible to work with; not always of course; pure set theory works happily with infinities. – Mozibur Ullah Sep 24 '14 at 0:38
  • The first cardinality is the cardinal of the number of points in the compact interval; or forgetting its points its locale of opens. Is there something similar for the higher infinites? – Mozibur Ullah Sep 24 '14 at 0:41
  • Might get more takers if asked on mathematics.se since the question is specifically more number/set theoretic than philosophical per se, especially regarding higher orders of infinity. (Sorry, I understand your question, but don't have an answer although I suspect that anything >aleph0 probably is not compatible by nature. There's probably a theorem for this, but I also wonder if it delves deep into the continuum problem.) – Howard Pautz Sep 24 '14 at 1:13
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    @MoziburUllah Any ordinal can be given the order topology; and topological spaces have compactifications. But you'll have more fun with this on math.SE as already suggested. – user4894 Sep 24 '14 at 1:38
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Here is the short mathematical answer :

  • An ordinal is compact if and only if it is a successor ordinal.
  • A cardinal is compact if and only if it is finite.

Here we are assuming the natural "order topology" is used. The result for cardinals follows from the result for ordinals since an infinite cardinal must be a limit ordinal.

If λ is a limit ordinal, then λ+1 is a "one-point compactification" of λ which we might loosely express as :

  • λ = [ 0, λ )
  • λ+1 = [ 0, λ ]

borrowing our notation from that of intervals in the real line.

The notion of compactification you describe in your question is strongly Euclidean and so does not readily apply to the ordinals in general. However, because it is tempting to view the ordinals as a line in the plane, there are number of nifty pictorial representations of infinite ordinals, drawn in a compact region of the Euclidean plane. Here is one I found via google images of the limit ordinal ωω as a spiral :

Intuitively, this type of Euclidean pseudo compactification would only work for countable ordinals such as ωω, pictured here. One can (almost) imagine extending this picture to a higher countable ordinal by replacing each ordinal in this depiction with a copy of ωω to obtain something like ωω2 perhaps(?) (I'm not sure). Imagining further substitutions beyond that, things become less clear. The first uncountable ordinal ( ω1 ) would be at the "limit of the limit of the limit ..." of this process - if that makes any sense.

This sort of doodling is fun, but unfortunately it does not say anything about the formal, topological view of compactness among the ordinals.


EDIT : When one considers the class of all ordinals, compactification seems wholly inappropriate. Re-reading this post, I think that most of what I've said, after the initial mathematical statements, is waffle. Probably just a shameless excuse to squeeze in a pretty picture. The whole idea of closure is contrary to Cantor's formulation, as suggested by @Conifold 's comment below, even if it is well-defined from a topological point of view.

  • Somebody needs to make a spinning gif out of that. Nice writeup. – user4894 Sep 24 '14 at 22:46
  • @user4894 Ya, it's a great image. I want a spinning bow tie like that. Maybe with chrome fin accents. – Nick R Sep 24 '14 at 22:49
  • @Nick R I'd say that the idea behind ordinals and cardinals is the opposite to that of compactification. Cantor explicitly designed his "third generation principle" to break past the possibility of any "closure" for ordinals: "if already generated ordinals share a property then they can be grouped together to form a new ordinal" (which lacks the property). – Conifold Sep 25 '14 at 0:19
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    I agree - even with my modest understanding. I may have misunderstood the OP. – Nick R Sep 25 '14 at 1:46
  • Definitely a nice picture; it certainly seems 'possible' to make this picture rigorous at least upto epsilon_0 - the first uncountable ordinal; by 'embedding' them into the real line; which is more easily imagined by inserting a real interval between every two consecutive ordinals – Mozibur Ullah Sep 25 '14 at 17:16
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At least some ordinals and cardinals are for real. There is a notation for them such that you can write them down. But writing them down is not enough. You also want to be able to decide whether two different written down cardinals are equinumerous. For ordinals, you even want to compare them relative to their order, because the order is important for ordinals.

Now comes the problem. You can even write down the cardinal "1" in a way that it is nearly impossible to decide whether it is equal to "1" or not. But this is definitively not the fault of the cardinal "1". It's also not necessary a fault of the notation, because any sufficiently expressive notation will also allow to write down hard to decide expressions. (By hard to decide, I'm thinking of a computer which runs out of memory or time while trying to decide equinumerity.) For ordinals smaller than omega, writing them down in binary form makes it relatively easy to decide equinumerity, but converting such an ordinal from an arbitrary notation to this notation can increase the size of the description "more than" multi-exponential. But at least there is a notation which allows do decide equinumerity for these ordinals in principle. Such a notation also exists for significantly larger ordinals, but beyond a certain ordinal, no such notation exists any longer.

  • Interesting. 'You can even write down the cardinal "1" in a way that it is nearly impossible to decide whether it is equal to "1" or not'. I find this confusing - can you explain further. – Mozibur Ullah Sep 25 '14 at 14:49
  • @MoziburUllah Well, instead of "1", I could write down "53^(2*5^7)-(53^(5^7)+3^4-2^4*5)*(53^(5^7)-2^6+7*9)". Wolfram Alpha easily figured out that this is equal to "1". So I tried instead "5321234567^(2*5^71)-(5321234567^(5^71)+3^4-2^4*5)*(5321234567^(5^71)-2^6+7*9)", and now Wolfram Alpha told me this would be approximately "10^(10^(10^(1.706852871469356)))". So do you think that it is equal to "1" (and Wolfram Alpha is just too dumb to see this), or did I make a hidden typo somewhere, and Wolfram Alpha is right that this is different from "1"? – Thomas Klimpel Sep 25 '14 at 16:14
  • @:ah, ok; its the 'writing down' thats problmatic! Do you mean by 'nearly impossible' that there are formal systems where equality checking is undecidable? – Mozibur Ullah Sep 25 '14 at 16:56
  • @MoziburUllah By 'nearly impossible', I mean that equinumerity can't be decided within realistic memory and time bounds. An ordinal notation can be more restricted than a formal system for predicate logic, so that it can describe larger ordinals before it gets undecidable. In my example, I used subtraction to create problems, but typical ordinal notation systems won't include subtraction. But if they want to be able to describe large ordinals, then at some point they need to allow some problematic operations. – Thomas Klimpel Sep 25 '14 at 17:22
  • @ThomasKlimpel I can write down various formulations of Russel's paradox and lots of facts about unicorns. What does that have to do with being real? And who cares whether they are isomorphic under some set of rules that process those notations? I do not see how this is an answer. – jobermark Sep 25 '14 at 20:20
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In some sense, it is clear these things are idealization, but they capture human intuitions that simplify or organize notions of arbitrary complexity or continuation. So they are Platonically 'real'. The complexity they model is not necessarily geometric or topological, just about relations. So I am not sure why anyone cares about their compactness.

I would say that compactness is not what is primarily gained by closing the complex plane, the real line, etc. A great deal of intuitive value is added by adding various idealized numbers into computations. Consider the nonstandard analysis of Los, et. al. It allows the kind of 'Cauchy-style' arguments that calculus used early in its logical development. Physicists routinely use this style of argument, involving infinitesimals and meaningful integration over discontinuous deltas, to simplify their handling of calculations. Creating these extra 'things' lets us write down the rules that tell us when such intuitive arguments are trustworthy, and when they cross a line into meaninglessness.

The closure of the complex plane is not important primarily because it is compact, but because it gives a total symmetry to computations involving things like conformal flow. The artificial infinities of the reals are helpful in avoiding carefully tracking epsilons and deltas that will slowly drive you nuts. The artificial infinitesimals make it possible to think through more complex analytic problems with less work.

Ordinal and cardinal infinities play a different role, in facilitation things like transfinite induction. It can be quite hard to imagine handling all varieties of sets and tracking all edge cases, but the isomorphism of L or V to the whole category of sets leveraged through ordinal induction gives us a way to prove things about arbitrarily varied structures in a systematic way.

  • +1 for 'epsilons and deltas will slowly drive you nuts'; can't agree more ;). Infinitesimals aren't artificial - they're for real; – Mozibur Ullah Sep 25 '14 at 20:32
  • Well, a whole system that allows them doesn't work. Like all other forms of infinity, they have to be provisionally real, and carefully protected from creating immediate contradictions by reducing the power of the logic working on them. You cannot have absolute negation (in a pervasive theme for me) and infinitesimals. – jobermark Sep 25 '14 at 20:33
  • Sorry for the massive reedit right as you were posting. – jobermark Sep 25 '14 at 20:38
  • Isn't this true for everything? I just mean that theres a geometric notion of the infinitesimals which relies on constructive logic - is this a stronger or weaker logic in your book? You can't prove the law of the excluded middle in them. – Mozibur Ullah Sep 25 '14 at 20:39
  • It can prove less, so it is necessarily a weaker logic. But I am in the end a progressive constructivist. I am not going to say simple existence by reductio-ad-absurdum is not a proof. But I think that a result is really destined for the dustbin if it goes too many generations without a constructive proof. – jobermark Sep 25 '14 at 20:42

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