Talk:Large countable ordinal

message: 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)
thread_title: Untitled

message: 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 λk, we let:
A (λ, n) = A (λn+1, n+1).
Then A (ε0, 9!!!) would be a large finite number.
by JRSpriggs 07:37, 15 March 2006 (UTC)
thread_title:  Applying large ordinals to large natural numbers 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Title again 

message: I wrote this description of an ordinal collapsing function which provides a system of notations up to the Bachmann-Howard ordinal (with a note of how to go further).  It turned out to be longer than I expected, in fact (but the main reason for writing it was to clarify my own thoughts as to how ordinal collapsing functions work, and to provide a simpler vision of the Bachmann-Howard ordinal than the transcountable Veblen scheme of two variables).  Anyway, comments are welcome, especially concerning the question of which articles this could be merged into (if any) and how (and which parts are actually useful for Wikipedia).  One might accuse me of bordering on original research (horresco referens!) because the function in question is not exactly one which has been published already, but I think it is close enough to the ψ functions of Buchholz to ward me from that reproach (the reason for any changes is purely pedagogical).  Any thoughts?  --Gro-Tsen (talk) 16:26, 16 March 2008 (UTC)
thread_title:  An ordinal collapsing function 

message: Since nobody is responding to the above call for suggestions on where/how to insert my text on constructing an ordinal collapsing function, I'll make my own suggestion.  I propose having
* one article for "predicatively defined ordinals" (title is debatable, though, and probably disagreeable to those who insist the Feferman-Schütte ordinal is the largest predicative one), which would merge the Veblen functions (extended to finitely many, then transfinitely many variables — this is perhaps Schütte's Klammersymbol but I'm not sure), the (not very notable) Ackermann ordinal, and the small and large Veblen ordinals, and of course the Feferman-Schütte ordinal;
* another article for "ordinal collapsing" (or something), which would merge in my above-mentioned description, thus describing the Bachmann-Howard ordinal in detail, and then go on to mention some even larger proof-theoretic ordinals such as Rathjen's collapsing of a weakly compact cardinal.
How does this sound?  The idea of having one page for each remarkable named ordinal seemed nice at first but now I think merging them in two broad families would be better.  Of course, the large countable ordinal article would be slightly reorganized accordingly. --Gro-Tsen (talk) 10:15, 12 April 2008 (UTC)
replies: text:  WP:BOLD Zero sharp (talk) 03:13, 17 April 2008 (UTC)
thread_title:  Suggestions for merging 

message: I am interested in the quoted result that the ordinal strength of a system is equal to the first countable ordinal which is not proven to be a countable ordinal in the system. I could not figure out whether the strength of ZF would be this ordinal, or the possibly larger ordinal given by the limit of the images of some collapsing functions operating on the uncountable ordinals as well. I did not see any obvious reason that the two ordinals must be the same, even if the collapsing function is shown to be a function in ZF, but perhaps it is obvious. The claim here is that the "useless" definition of the ZF ordinal can prove the arithmetical consequences of ZF, but what is the statement of this theorem, and the literature source?Likebox (talk) 23:50, 28 October 2009 (UTC)
replies: text: Define the ordinal strength of a (true) theory as the least least ordinal which is not the order type of some definable binary relation on a subset of ω which can be proven to be a well-ordering in the given theory. See Ordinal analysis. JRSpriggs (talk) 10:11, 29 October 2009 (UTC)
replies: text:  Where is this result in the literature? I just wanted to learn more about it.Likebox (talk) 15:22, 29 October 2009 (UTC)
replies: text:  Since you are giving a definition, by result I mean the following: if you know the ordinal strength of the theory, you can know which recursive function growth-rate is not provably recursive, and further that for theories with faster growth rates the ordinal is bigger.Likebox (talk) 15:27, 29 October 2009 (UTC)
replies: text:  Also, given a theory T with ordinal a, what is the ordinal of T+consis(T)?Likebox (talk) 15:36, 29 October 2009 (UTC)
replies: text:  (For posterity, since this is an old discussion) If T is a consistent, primitive recursive, Σ11-sound theory that extends PA, T+Con(T) has the same proof theoretic ordinal as T. In general, adding any true Σ11 sentence to T does not change its proof theoretic ordinal, since we assumed T is consistent, Con(T) is such a sentence. (Proposition 2.24 in Rathjen's The Realm of Ordinal Analysis.) C7XWiki (talk) 08:32, 4 December 2022 (UTC)
thread_title:  Literature pointer 

message: The article contains a questionable assertion, "Likewise, the least-fixed-point operator used in the Veblen hierarchy is not predicative.", which it cites to Nik Weaver's paper "Predicativity beyond Gamma_0".  Weaver is a skeptic of the Feferman-Schutte analysis of predicative second-order arithmetic, which places the Feferman-Schutte ordinal Gamma_0 as the upper bound of the predicative ordinals, and his paper instead argues that the bound for predicativity should be placed much lower, because he thinks that the Veblen hierarchy involves impredicative definitions.  This viewpoint has pretty much no mainstream acceptance, so I think this sentence may need to be removed, and perhaps even the previous sentence "This definition is impredicative, because it uses the uncountable ordinal Ω, which, in some sense, already uses all the countable ordinals we are trying to construct in its construction.".  (I don't know whether this assertion has mainstream acceptability or not, but in any case it would be better to source it to someone other than Weaver.) 69.248.132.69 (talk) 02:55, 8 December 2013 (UTC)
replies: text: As far as I'm concerned, the word "predicative" is at best a vague convention (ordinals whose definition somehow does not refer to larger ordinals: which, obviously, depends very much on what we mean by "definition") or an ad hoc definition (defined to be exactly ordinals less than Γ₀), and at worse philosophical mumbo-jumbo.  I don't even understand what the debate is about (you say Weaver argues that the predicativity bound should be placed lower than Γ₀ whereas his paper is called "predicativity beyond Γ₀"?  is that right?), nor do I understand what's magical about Γ₀.  So anyway, as far as I'm concerned, you can edit this as you like, and if you can somehow explain to me that predicativity is not a useless or ill-defined notion, all for the better!  --Gro-Tsen (talk) 21:42, 8 December 2013 (UTC)
thread_title:  Questionable assertion about Predicativity 

message: Even though the functions are described (and perhaps proven by others) to be sophisticated, since the research is self-published it may not be eligible for Wikipedia C7XWiki (talk) 20:08, 11 September 2021 (UTC)
thread_title:  Maksudov's functions