Talk:Goodstein's theorem

message: Both this article and the article for Paris-Harrington theorem claim to be the first natural theorems not provable by PA.  I don't know which is true, but I suspect they can't both be true at the same time.  165.125.144.16 (talk) 20:35, 13 April 2009 (UTC)
thread_title:  Two Firsts 

message: The natural theorem expressible but not provable in PA is Gentzen's induction principle. He maks this very clear in 1943 and it has been noted many times, e.g., in Schwichtenberg's paper in the Handbook of Mathematical Logic, ed. Barwise. 1977.  —Preceding unsigned comment added by SternJacob (talk • contribs) 13:54, 24 May 2009 (UTC)
thread_title:  One First 

message: Is there some analysis of the ordinal strength of this theorem?  It looks to me like it follows from induction on \epsilon_0, which makes it equivalent to CON(PA), but the article doesn't come out and say so, and I might be missing something.  Could someone knowledgeable update the article one way or another?  Thanks. 67.122.209.126 (talk) 12:03, 1 June 2009 (UTC)
thread_title:  ordinal strength 

message: "for example,
\omega^{\omega^\omega} + \omega decreases to
\omega^{\omega^\omega} + 4" - well I somewhat get that. However, can someone please describe what happens when we decrease \omega^{\omega^\omega} + \omega? --Hirak 99 (talk) 19:49, 2 November 2009 (UTC)
replies: text: If you start with 4^{4^4}+4, which corresponds to \omega^{\omega^\omega}+\omega, the next element in the Goodstein sequence is 5^{5^5}+5 - 1 = 5^{5^5} + 4, which corresponds to \omega^{\omega^\omega}+4 because only the 5s are replaced by ωs. If this is not what you are asking, could you rephrase your question? — Carl (CBM · talk) 19:57, 2 November 2009 (UTC)
thread_title:  Question on Proof 

message: The Hydra game appears to be a derivation of Sprouts (game).  —Preceding unsigned comment added by Dexter Nextnumber (talk • contribs) 22:46, 9 January 2010 (UTC)
thread_title:  Connected to Sprouts? 

message: The article had contained the following two external links (the second one, recently added, being to a page of my own creation):
* The Hydra game implemented as a Java applet
* Generalised Hydra game as a bracket-expression rewriting system 
A user removed the second link (but left the first one), commenting that it was "not directly related".  However, trees can be represented using either graphs (as at the first link) or by using bracket expressions (as at the second link).  The relationship to Goodstein's theorem is exactly the same for both representations of the Hydra game, so I suggest a more evenhanded treatment.  The fact that the second link presents the game as the execution of a "program" composed of trees, and also explains a more general form of the game, would hardly seem to matter in this regard.  On the other hand, if the concern is really about academic or other credentials (rather than relevancy), then I don't wish to pursue the matter.  Discussion? — r.e.s. (talk)  12:53, 8 April 2010 (UTC)
replies: text: I lean toward removing both, actually, but you really shouldn't add a link to your own research, even though your academic credentials are not being questioned—it's not a matter of academic credentials, but of WP:EL#ADV.  I think relevancy may have been the wrong reason for deletion, and I apologize for not checking more carefully.  I'll to consider this more closely, and determine whether your page is the better one; if so, I'll restore it.    — Arthur Rubin  (talk) 13:53, 8 April 2010 (UTC)
replies: text: Perhaps it would be best to have a WP page on Hydra games; but, lacking that, I hope at least one of these links is retained. If it's a choice between the two, I recommend the first link, since pictorial graphs are the "standard" and are probably visually preferable.  Aside from some generalisations, the second link primarily shows how these games can be played using bracket expressions rather than graphs. — r.e.s. (talk)  14:26, 8 April 2010 (UTC)
thread_title: Hydra games

message: The lede is carefully worded to avoid giving the impression that the result can be "proved" in PA but as a result the lede never actually states that Goodstein actually proved anything, only "stated".  Should this be clarified?  Tkuvho (talk) 11:53, 14 May 2012 (UTC)
replies: text: Yeah, that's silly.  The easiest solution is just to say "proved", and then explain further down how much strength you need to prove it, and that that's more than PA. --Trovatore (talk) 19:15, 14 May 2012 (UTC)
thread_title:  a statement made? 

message: Long time no edit …
I converted many of the inline < math> tags with no more than one level of super/subscript to inline HTML, and corrected the italicization of such phrases as G(m)(n) to G(m)(n).  (This looks more clear when you edit it in WikiText mode; using the Visual Editor does not see the difference.)  If anyone objects, please let me know.  — Arthur Rubin  (talk) 23:23, 21 June 2015 (UTC)
thread_title:  Reformatting 

message: The whole para “Then f(G(m)(n), n+1) = f(G'(m)(n), n+2). Now we apply the minus 1 operation, and f(G'(m)(n),n+2) > f(G(m)(n+1),n+2), as G'(m)(n)=G(m)(n+1)+1.  […] is not an ordinal.” is hard to understand. Could please someone rewrite it starting with simple steps and ending with conclusions? Already the second sentence is not understandable, and the whole para (in fact, the whole claim) needs a complete clean rewrite from scratch. A so-called “example” is not helpful the way it is written. Even if it were, an example is not a proof. — Preceding unsigned comment added by PeterMuellerr0 (talk • contribs) 20:51, 2 February 2024 (UTC)
replies: text: I don't think it's that bad. Apparently the first sentence is considered self-evident since it just says replacing the base n+1 by n+2 and then n+2 by ω is equivalent to directly replacing n+1 by ω. The second sentence makes use of the fact that  is strictly monotone in u (which is maybe not that obvious, but at least somewhat intuitive), and the part after "as" is just the observation  from the last paragraph stated in reverse. It probably could be better, but it's not that bad. Bbbbbbbbba (talk) 05:56, 30 May 2024 (UTC)
thread_title:  Bad proof