Talk:Transfinite induction

message: Currently, transfinite recursion is a redirect here.  That's not necessarily bad.  But if it's to stay that way, then the page really ought to say something about transfinite recursion (a method of definition, not of proof). --Trovatore 01:51, 16 July 2005 (UTC)
replies: text: First cut in place.  Could use an example or two, and possibly some a more rigorous treatment in addition to (not instead of) the informal descriptions given. --Trovatore 07:03, 16 July 2005 (UTC)
thread_title:  Transfinite recursion 

message: The page says "Notice that the second and third cases are identical except for the type of ordinal considered (...)",  talking about the Successor Case and Limit Case of Transfinite induction. This doesn't seem right; the cases are different because in the Limit case, the limit ordinal is not in the form β+1. I think this comment is misleading (at best), and should be removed. Any objections? --RRM 16:39, 4 April 2007 (UTC)
replies: text: The similarity lies in the fact that one is allowed, in both cases, to use the property for all previous ordinals in showing it for the current ordinal. So a common proof is often possible. JRSpriggs 07:47, 5 April 2007 (UTC)
thread_title:  Successor Case vs Limit Case 

message: Ordinals form class, not set. —The preceding unsigned comment was added by 83.5.233.94 (talk) 11:05, 14 April 2007 (UTC)
replies: text: The class of all ordinals is not a set. That doesn't mean there's no such thing as a set of ordinals. --Trovatore 18:33, 15 April 2007 (UTC)
thread_title:  Sets of ordinals??? 

message: What?--68.161.161.206 (talk) 11:52, 18 April 2008 (UTC)
replies: text: The first and last order signs should actually be angle brackets, meaning the sequence of elements rα indexed by α < c. — EJ (talk) 12:25, 18 April 2008 (UTC)
thread_title:  " <''r''<sub>α</sub> | α<c >" 

message: Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all  β < α, then P(α) is also true.  Then transfinite induction tells us that P is true for all ordinals.
That is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.
But induction is valid only if the property P(α) is true for minimal element: α=0. I quess it should be mentioned.
Eugepros (talk) 06:58, 10 August 2010 (UTC)
replies: text: If there are no smaller elements, then they all satisfy P vacuously. Thus you get P(0) from nothing and the induction begins. JRSpriggs (talk) 07:20, 10 August 2010 (UTC)
thread_title:  How about minimal element? 

message: The lead section talks about well-ordered sets, the rest of the article about ordinals.  That is not nice.  — Preceding unsigned comment added by 24.85.86.45 (talk) 04:55, 23 October 2011 (UTC)
replies: text: Ordinals are the archetypical well-ordered sets. Every well-ordered set is order-isomorphic to an ordinal, and ordinals are well-ordered sets. So there should be no problem. JRSpriggs (talk) 10:23, 23 October 2011 (UTC)
thread_title:  well-ordered vs. ordinals 

message: The example states that choice is only used to well-order the reals. But the actual "construction" doesn't seem a real "construction". Specifically, the claim to be able to just "countinue; at each step" to chose uncountably often does not seem well-defined at all. Of course it is possible to clearly state this process in a formally correct way, but then it won't demonstrate what is meant to be demonstrated here. 217.84.235.60 (talk) 22:50, 11 January 2012 (UTC)
replies: text: I am not sure I understand the objection.  As far as I can tell, once the original wellorder is chosen, after that everything is completely well-specified and constructive; at each step you choose the least possible element of the equivalence class, "least" being understood in the sense of the chosen wellorder.  Are you claiming that the iteration as stated can proceed in two different ways, even given the same wellorder?  I think that that is not true. --Trovatore (talk) 23:28, 11 January 2012 (UTC)
replies: text: To clarify a bit: The reason this example is in the article about transfinite induction is, presumably, to demonstrate it in use. But precisely this use is totally obscured in "chose at each step". For the reader who is not clear on what transfinite induction is, its use in that example is not at all obvious. More dangerously, these matters need very precise attention especially when in a learning stage. But looking anywhere in the article, nowhere is "chose something" uncountably often mentioned - instead, either a restriction of a class function (no strict classes mentioned here), or a version with precise steps to follow regarding the treatment of successor and limit ordinals. Neither of those appear anywhere either. I would think that any example on this page should take great care to be exact and rigorous in its use of mathematical terms, and this section is anything but.217.84.235.60 (talk) 05:03, 12 January 2012 (UTC)
replies: text: I still don't follow you.  Is your objection that what happens at each step is not actually "choosing", because it's pre-determined by the chosen wellorder?  Would you be happier if the word choose were replaced by take or some such?  As for proper classes, there are none in sight, so I don't see the point of mentioning them. --Trovatore (talk) 05:11, 12 January 2012 (UTC)
replies: text:  Maybe what you're really looking for is some sort of explanation as to how to turn the argument into a formal deduction from ZFC or something?  In that case I completely disagree.  The important thing to get across here is what's really going on, "naively" or "Platonistically" if you will, and what's really going on is an uncountable sequence of discrete steps.  There is nothing ill-specified about it; it's completely canonical.  The details of how to create a formal ZFC argument are less canonical, and also less interesting. --Trovatore (talk) 06:30, 12 January 2012 (UTC)
replies: text: This last bit is a bit closer, but not fully. I don't mean a formal deduction from ZFC - that'd certainly be far over the top. But a somewhat more formal deduction from the principles actually laid out in the paragraph above is what I'd expect to see, or at least hinting at it. The transition from having functions G as used in the above explanation to "doing something in steps" is what is needed to really "understand" transfinite induction, yet it is - for the most likely reader of this page, i.e. someone not yet familiar with how these things work - not trivial. Adding something like "so we have G defined as ... and then we apply the principles outlayed above" would help. To be clear: I'm not disputing the correctness of the statements there. I just don't think the example is particularly serving its purpose of showing where AoC is used and where the transfinite induction happens (and that AoC is not used there any more).217.84.242.179 (talk) 16:10, 12 January 2012 (UTC)
replies: text: Actually, a fine example of what I'd expect to be explained is here: http://en.wikipedia.org/wiki/Ordinal_number#Transfinite_induction - that the main page has a less clear description of what's going on in the only example given is what I was trying to point out.  — Preceding unsigned comment added by 217.84.234.165 (talk) 21:53, 14 January 2012 (UTC)
thread_title:  Vitali set example problematic? 

message: I'm not happy with the footnotes.  The second one is not entirely bad, but the first one assumes ZF set theory.  In NBG or MK set theories (among others), a "class function" is just a function which is not necessarily a "set".  Also, transfinite recursion doesn't quite work in ZF.  You can correct the conclusion as follows:
#(Traditional) for any ordinal λ there is a function F: λ → V such that for all α < λ, ....
#(Advanced)  There is a "class function" F: Ord → V such that ....
This article needs specific references.  I don't have a copy of Suppes's book.  I do have a copy of my late mother's book, Set Theory for the Mathematician, which has most of the relevant theorems, but uses NBG or MK as the underlying set theory. (I also have Kelley's book, General Topology, but I don't think it actual uses transfinite induction.  It also uses, of course, Morse-Kelley set theory, rather than ZF.) — Arthur Rubin  (talk) 07:50, 23 August 2015 (UTC)
thread_title:  Problems (2015) 

message: I just noticed that some time ago a section header titled "Transfinite induction" was removed by a bot, effectively moving the contents of the first section into the lede section, see . I think the reason for changes like this is that the section title was equal to the page's title, leading to conflicting  and  IDs, see MOS:HEAD: "Section and subsection headings should preferably be unique within a page; otherwise section links may lead to the wrong place, and automatic edit summaries can be ambiguous." Anyway, I think I actually prefer the flow of the page before this change, with two sections "Transfinite induction" and "Transfinite recursion", see . What do the other wikipedians say? Should something like the section title be reinstated? – Tobias Bergemann (talk) 10:37, 8 October 2015 (UTC)
thread_title:  Sections "Transfinite induction" and "Transfinite recursion"