Talk:Large countable ordinal

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I created this article by extracting a section from the article on "ordinal number". See the discussion for that article on the creation of this one. JRSpriggs 08:23, 13 March 2006 (UTC)

Contents 1 Title of article 2 Applying large ordinals to large natural numbers 3 Introduction 4 Title again

[edit] Title of article

The article looks great (not that I've looked at the details). But the title's gotta go. The ordinals in question might be large from a proof theorist's point of view, but not from that of a set theorist—all the ones mentioned except for in one brief sentence are below ω₁.

If the article is to be primarily about proof-theoretic ordinals, then that might be the title (except it should be singular: "Proof-theoretic ordinal"). Another possibility might be List of named ordinals, except it's not in list format currently and I'm not sure it would be an improvement to put it in that format. Specific countable ordinals? Named countable ordinals? Maybe just Countable ordinal; that would give a rationale for this text to be separate from ordinal and give space for all this discussion. At the moment I think I like the last suggestion best, but others are solicited. --Trovatore 17:16, 13 March 2006 (UTC)

Whether an ordinal is large or not depends on one's point of view. To someone used to thinking in terms of large cardinals, these are small. But to someone who is only thinking about ordinals with constructive notations, these are large. Tell me, are there any ordinals which are large in your sense and are discussed in the literature other than the initial ordinals of large cardinals? I think not. JRSpriggs 08:08, 14 March 2006 (UTC)

I agree with JRSpriggs's comment above. Perhaps the title is not the best possible, but I can't think of better, and the argument that large ordinals should be large cardinals doesn't strike me as convincing at all. --Gro-Tsen 09:06, 14 March 2006 (UTC)

Well, of course it depends on point of view. That's part of what makes "large ordinals" a bad title. I didn't say large ordinals should be large cardinals; I'm not suggesting a different subject matter under the title "large ordinals" (which as far as I know is not a standard term, so we shouldn't have an article called that, no matter the subject matter) but rather a different title for this article.

I don't claim I have a really good title for this article, but I have mentioned several better ones. Please discuss your objections to these. --Trovatore 14:56, 14 March 2006 (UTC)

Well, it's obvious, isn't it? Now suppose I wish to also say a few things about large ordinals which aren't necessarily countable. E.g., the smallest α such that V_α is a model of ZFC: should I create another article, or what? The way I defined admissible ordinals, they don't have to be countable: so discussing them in full generality os now off-topic. You could at least have waited for some sort of consensus here, or at least for the discussion to settle down, before moving the article! --Gro-Tsen 18:17, 15 March 2006 (UTC)

Granted, it might not have been the most politic move. But no one was responding, and the title was horrible. The α you mention above is countable, of course (that's not provable in ZFC, but then neither is its existence). Admissible ordinals should have an article of their own, and I suspect anything you'd want to say about uncountable ordinals would be sufficiently different in character from this article that it wouldn't be naturally included here. --Trovatore 20:39, 15 March 2006 (UTC)

No, the smallest α such that "V_α is a model of ZFC" is not countable (indeed, "being countable" is absolute for all V_α with α>ω limit (or some such thing) because whenever X is a countable set of rank less than α any bijection with ω will also have rank less than α; so the existence of ω₁ being a theorem of ZF precludes any V_α with countable α from being a model of ZFC). It has countable cofinality, however (by Skolem closure), and it is less than the first inaccessible. All of this might have been worth mentioning in the article (since passing note is made of the first α such that "L_α is a model of ZFC" — which is countable), except that it doesn't fit the title. As for the "no one was responding", couldn't you have waited even one week? --Gro-Tsen 23:11, 16 March 2006 (UTC)

Yeah, you're right on points 1 and 3. On point 2, including that α in this article, well, this is probably not the best final title. What the ordinals discussed have in common is not so much being "large" but rather having "names" or, if you like, sufficiently absolute definitions. "Large" should probably not be a part of the name the article winds up under. While I doubt it's particularly interesting to mention that particular α, we can probably choose a name that accommodates it if you have your heart set on including it. --Trovatore 23:35, 16 March 2006 (UTC)

[edit] Applying large ordinals to large natural numbers

We have seen that large uncountable ordinals can help one to describe large countable ordinals. Similarly, large countable ordinals can help one to describe large finite ordinals (natural numbers).

For example, one can extend the Ackermann function to the transfinite as follows:

A (0, n) = n+1.

A (α+1, 0) = A (α, 1).

A (α+1, n+1) = A (α, A (α+1, n)).

If λ is a limit ordinal with the fundamental sequence which takes k to λ_k, we let:

A (λ, n) = A (λ_n+1, n+1).

Then A (ε₀, 9!!!) would be a large finite number.

by JRSpriggs 07:37, 15 March 2006 (UTC)

[edit] Introduction

Per WP:LEAD and Wikipedia:Guide to writing better articles#Introductory material, it is not correct to have the article start with a section called "Introduction" and no text before the first section header. It's also not great style to talk about "this article" (think about articles in a print encyclopedia; when have you ever seen that there?). Moreover there is no reason to tell people not to look here for information about large cardinals, as there is no reason to think anyone would do so, and the line

The ordinals described here are large only in relationship to ordinals which have contructive notations (descriptions).

is misleading, given that a large fraction of the article is devoted to ordinals that do have constructive notations. The "Introduction" header should be removed, along with the self-referential text about "this article". --Trovatore 23:44, 16 March 2006 (UTC)

I hope that these problems have been adequately fixed now. JRSpriggs 06:35, 20 March 2006 (UTC)

To be honest I still don't like the first two sentences, referring to "the reader" and to the content of another article. This sort of "metalanguage" is, in my opinion, poor style except in things like disambiguation links, though there are times it's hard to get around. But I don't think it's hard to get around here, as neither sentence is really necessary: the reader can find what he needs through prominent links in the article, and the thing about large cardinals was always a red herring. You were the one who brought that up, and I never understood why.

Part of the problem is that it's still not a good article title. We should find one without the word "large", which is not really what distinguishes these ordinals. What distinguishes them is that they fall into certain naming schemes, or perhaps that they have sufficiently "absolute" definitions (under forcing, say). --Trovatore 15:23, 20 March 2006 (UTC)

Although I understand your dislike of "metalanguage", I feel that the reference to "ordinal arithmetic" is necessary. Ordinal arithmetic is used in this article, especially in the two subsections on the Veblen hierarchy. But there is no obvious place to work a reference to "ordinal arithmetic" into the language which would give the reader a sufficient nudge to get him to look at the other article. If you can figure out how to do it, please try. My reference to "large cardinals" was an attempt to cope with the confusion (which you pointed out) which a reader might be led into by the title "LARGE countable ordinals"; and the remainder of that sentence is an attempt to justify why I used the word "large" inspite of the fact that these ordinals are only countable. These ordinals ARE large compared to the other ordinals discussed in "ordinal number" and "ordinal arithmetic" which are generally less than or equal to epsilon-zero. The final sentence of that paragraph, "Larger and larger ordinals can be defined, but they become more and more difficult to describe.", summarizes the main point of this article. JRSpriggs 06:47, 21 March 2006 (UTC)

It just isn't standard practice. There are lots and lots of articles that can't really be understood by someone without prior knowledge (probably almost all the articles in the math project, really) and we don't try to do this sort of pointing (though there have been proposals, such as using templates; that would be ugly in my opinion but would not raise the "metalanguage" problem). I don't really agree with the last claim starting "these ordinals ARE large", because while it's true with respect to the ordinals discussed "retail" in those articles, the articles are really more about ordinals "wholesale", named or not. And the ordinals discussed in this article are not large compared to most of those.

So really, it's still not a good name for the article. Once we get it to a better title, many of the problems you raise will cease to be problems. --Trovatore 07:07, 21 March 2006 (UTC)

[edit] Title again

Since Trovatore is still not happy with the title, I was thinking that something more along the line of "Constructive ordinal notations" might be good, provided that we add more material on Kleene's O and Takeuti's ordinal diagrams, etc.. JRSpriggs 10:33, 22 March 2006 (UTC)

I, for my part, would be even less happy: that would make the whole section "Beyond recursive ordinals" off-topic, and I'm waiting for someone more knowledgeable than I am to say interesting things about the hyperjump, recursively inaccessibles, recursively mahlos and so on (and there's another word I forget: "uncompressible" or something?). --Gro-Tsen 16:59, 22 March 2006 (UTC)

I agree; that's not the right title either. Something like "named ordinals", maybe. Although I think the stuff Gro-Tsen mentions should perhaps be a different article altogether—it seems to be about large-cardinal-like properties that can be reflected into the countable hierarchy; that's quite different in character from the subject matter of the current article. --Trovatore 17:09, 22 March 2006 (UTC)

I don't want to step on any toes regarding the title —I have no answers to the problems y'all have been discussing in that regard— but I do want to make the title conform to Wikipedia:Naming conventions (plurals). So I'm going to move it to Large countable ordinal and fix the double redirects, but please don't interpret that as my support (or opposition) to that title (over Large ordinal, Proof-theoretic ordinal, etc). —Toby Bartels 01:36, 9 September 2006 (UTC)