Talk:Ordinal number/Archive 4

Replace order type by order isomorphic?
Although I wrote the initial sentence (an ordinal is the order type of a well-ordered set), I'm now having the following misgiving about it. Further down we mention the historical definition in terms of equivalence classes and say that this must be abandoned in ZFC. But the order type concept is intrinsically based on equivalence classes. The alert reader might find this bothersome.

One solution would be to preface the order-type definition with a suitable rider indicating that this definition is (pick one) intuitive, historical, or traditional, and then refer to the next one(s) as the modern definition(s). However the order-type definition is not really that intuitive anyway for people that don't routinely work with posets up to isomorphism (which readers would know that there are exactly two posets with two elements?), requiring further explanation in the lead to bring them up to speed on a concept that later on is going to be dismissed as outdated anyway. (I used it because I routinely work with abstract posets and this definition is more intuitive for me personally than any other, as it may well be to the other editors here for all I know, which would explain why no one's raised this issue before. Having had time to absorb the lessons of 9/11 I'm less evangelical about my world view than I used to be in all respects except evangelizing Alan Sokal to be less evangelical.)

I'm presently leaning towards starting out with one of the three definitions we say later on is equivalent to "ordinal". This will mean nothing to most of course, but I'm now persuaded that approximately the same is true of "order type." I do think we should start with a precise definition, but it should be as short as possible so as at least to create the appearance of understandability ("if I can just grasp the implications of those two words..."), and we should immediately give examples that build up intuition, perhaps based on the von Neumann definition, and make the point that no two ordinals are order isomorphic (even when they are isomorphic as sets, a detail that has confused some on this talk page), which conveys all the information that "order type" does without getting entangled in the much higher order concept of an equivalence class.

Sooner rather than later in the body we should go back to the definition in the first sentence and relate it to the von Neumann definition, pointing out that the evident fact that ordinals consist of ordinals is captured in the single word "hereditarily," and so on. We should also point out that (even without regularity) the finite ordinals are exactly the finite transitive sets, since (unless I'm confused) trichotomy follows by induction (is there a name for the smallest ordinal for which trichotomy does not follow by induction?). We should either argue how this works for each of 0,1,2 separately (probably not 3) or give the whole proof at once, whichever seems better; the former may be easier to follow. Also make the point that trichotomy is only needed to accommodate very large ordinals (wherever "very large" kicks in, my bad for not knowing). --Vaughan Pratt (talk) 07:34, 21 December 2008 (UTC)

Following up on the foregoing, I've written a new lead for the article. Please comment either way here. If and when some consensus emerges I'll either replace the current lead with the new one or delete the new one. --Vaughan Pratt (talk) 06:21, 23 December 2008 (UTC)


 * Haven't looked at it yet (by the way it probably shouldn't be in mainspace -- it would be better for you to move it to User:Vaughan Pratt/New lead for ordinal number). But for whatever it's worth I'd just like to say that an excellent intuitive introduction to the ordinals is in the Birthday Cantatatata fantasy in Gödel, Escher, Bach.  In fact it was that that convinced me that there was an intuitive (i.e. non-formal but still precise) notion of ordinal number. --Trovatore (talk) 07:01, 23 December 2008 (UTC)


 * Oh, also: I definitely don't think it's a good idea to write the article as though "ordinal" is synonymous with "von Neumann ordinal".  The von Neumann ordinals are as I see it a (remarkably convenient, but somewhat arbitrary) way of coding the ordinals as sets; they're not what ordinals are. --Trovatore (talk) 07:14, 23 December 2008 (UTC)
 * Oops, I screwed up. Usually I do put these candidates in my sandbox, but forgot to do so in this case.  Thanks for pointing this out.
 * Should we put Hofstadter's account in the lead, or refer to it in the body?
 * Regarding what the ordinals "are", are you in the Wikipedia category "alert reader" referred to in the first paragraph of this section? Your view of them as an order type is dismissed at Ordinal_number as a now marginalized viewpoint inconsistent with ZF.  I happen to like that view myself, which is why I put in the first sentence a while back (I'm in the "very slow to be alert" category).  However the conception of an ordinal as a transitive set of ordinals has the distinct advantage of raising no foundational questions such as, what does it mean for ω to be a class?  Intuitively we should know what a class is, which is presumably where Russell was coming from, but that's old-school foundations that have gone the way of Principia Mathematica.  The modern definition of an ordinal is a hereditarily transitive set, which is to say, a transitive set of ordinals.  This is not von Neumann's definition, and nowhere in my proposed lead is von Neumann mentioned (he belongs as a historical footnote in the body, having failed to come up with the transitive-set definition), but remarkably the two definitions turn out to be completely equivalent (the point of the two "remarkably"'s in my candidate lead, as well as the part where I said "only one set of each finite cardinality can be transitive," whose significance is admittedly easily overlooked).
 * If you disagree I would be very pleased to see your example of an element of a model of ZF that is a finite transitive set yet not a von Neumann ordinal. In light of the definition of an ordinal as a hereditarily transitive set I think you will find that there is nothing arbitrary about von Neumann's encoding, though I imagine von Neumann himself would have taken your side on that!  As would have I before I saw what others had been getting at here, but perhaps not communicating as clearly as they might have.  (The only entities I hate more than mathematicians are terrorists like Alan Sokal and computers.  Dentists are fine, even cats are ok if you're patient with them.  Perhaps I'm being unfair to mathematicians, I should group them with cats.)  --Vaughan Pratt (talk) 05:57, 24 December 2008 (UTC)


 * Vaughan, I'm not going to attempt a point-by-point reply — it's too frustrating; you always throw out so many points.
 * Here's what I'm saying: I don't think we should say ordinals are hereditarily transitive sets, or even that they're order types viewed as equivalence classes. These are simply ways of coding the notion of ordinal so we can treat ordinals as sets.  (I also disagree, by the way, that order types are equivalence classes — that's just another coding.)
 * But ordinals are not sets, not really, though . Cantor dealt with ordinals without treating them as sets at all.  There was nothing wrong with his approach here.  (There is something wrong, as we know, with assuming that all the ordinals can be gathered together into a completed totality, but that's a separate issue.)
 * So what I'm saying is, we should treat ordinals more or less as Cantor did (though of course using modern terminology rather than his awkward "numbers of the first class" and so on). Then we should explain how they can be coded as sets.
 * Really this isn't particularly different from how we treat the natural numbers. No one, I hope, really thinks that the number 3 is {0,{0},{0,{0}}}, nor that it's the class of all classes with three elements — again, these are just codings; we understand what the Platonic 3 is at a much deeper level, one that precedes all our formalisms.  And the same is true of &omega;+1, though it's an intuition that takes longer to develop. --Trovatore (talk) 08:42, 24 December 2008 (UTC)
 * Sorry about the points---I should have just asked what you meant, which you explained just now.
 * But then are you suggesting that the ordinal article replace definitions with intuitions? If the article began "In philosophy" I'd be ok with that.  If you're ok with intuitions in place of definitions in a mathematics article then we could consider changing the first sentence of Group (mathematics) to "In mathematics, a group is a monoid whose actions are invertible."  --Vaughan Pratt (talk) 09:00, 24 December 2008 (UTC)
 * Let me put it this way. In the opening to natural number we don't define how naturals are coded as sets; we just explain what they are.  I think that's the direction we should go, at the top of the article.  Of course the situation is a little different here, because everyone already has a concept corresponding to the naturals, whereas many readers will need to create their concept of the ordinals.  But it isn't that different.  The ordinals we should be trying to explain, at the start, are Cantor's ordinals, not von Neumann's.  Yes, of course they're really the same, but for Cantor's ordinals the representation is simply left unspecified, and I think that's the way to go in the lede. --Trovatore (talk) 19:18, 24 December 2008 (UTC)
 * Well, you're the one working in set theory, not me. If you tell me that "hereditarily transitive set" is only a representation for ordinals and not the actual definition I'll take your word for it (unless a horde of set theorists jumps on me and tells me otherwise).  But in that case the article is worded misleadingly, and the distinction needs to be made clear.  I propose to amend the statement where it misleadingly says an ordinal is a hereditarily transitive set and correct it to say that an ordinal can be represented as a hereditarily transitive set.  This is an important distinction that needs to be made clear for any mathematical object.  Is that ok with you?   --Vaughan Pratt (talk) 22:30, 24 December 2008 (UTC)
 * That's certainly the solution I would prefer. Of course this preference is colored by my frankly Platonist POV.  If there are objections from further "left" on the spectrum then it may be necessary to fin a compromise. --Trovatore (talk) 00:35, 25 December 2008 (UTC)
 * Did Plato identify things with classes, or is that how modern Platonists define things? My preference is to define mathematical objects in terms of what properties they have, e.g. an ordinal is a set (if you're happy with defining an ordinal to be a class then I'm not clear as to why defining it to be a set is any worse) having the property of being hereditarily transitive.  Is that inconsistent with Platonism, and if so is there a name for that approach (to distinguish it from Platonism)?  --Vaughan Pratt (talk) 02:00, 25 December 2008 (UTC)
 * I fear we're drifting again, but OK, it's interesting. I'm using Platonist imprecisely; I really mean realist.  I don't see ordinals as being either sets or classes; to me they are sui generis abstract objects that logically precede sets.  Oh, they're not necessarily prior to sets of atoms, but prior to sets in the sense that we think of them in set theory; that is, objects that show up at some point in the iterative hierarchy.  In order to understand what a set is in that sense, you need to understand ordinals first, so it would be kind of circular if ordinals were sets.
 * Then, after the fact, you notice that ordinals can be represented as sets, and that this is a useful thing to do for all sorts of reasons, and from then on you normally don't bother to distinguish between abstract ordinals and ordinals represented as sets. --Trovatore (talk) 02:11, 25 December 2008 (UTC)
 * Very interesting. By a remarkable coincidence I was just writing today about a class of objects logically preceding sets, in that sets could be defined quite directly in terms of them.  They weren't motivated by that however, but by Euclidean geometry.  I figured no one would ever be interested in this other application of them (as a mathematical structure from which sets could be defined) because sets were already so entrenched in the mathematical consciousness, and so I dismissed that application out of hand.  Perhaps that was premature, maybe there is an audience for that point of view after all.  Though for you it would be competing with ordinals.
 * But now that I understand where you're coming from on this, it seems clear that you're a Russellian, thinking about sets the way Russell did, in terms of a type hierarchy. (Consistent with realism, in fact I guess you'd have to be a realist to accept ordinals sui generis.)  So then do you postulate the ordinals in order to rank the sets (by birthday as Conway would put it), where the rank of a set is defined to be an ordinal?  If so, do others at UCLA and Toronto view sets similarly today?  Is ωω 1 a perfectly fine rank for a set?  --Vaughan Pratt (talk) 03:53, 25 December 2008 (UTC)

Trovatore, after two months I've come round to your point of view that the von Neumann ordinals are a coding of the ordinals. The reason I was reluctant to call them a coding before was because the coding is equivalent to defining the ordinals as the hereditarily transitive sets. In fact in my proposed replacement for the lead I defined the ordinals as the hereditarily transitive sets, and if I hadn't actually exhibited the first few such you might not have complained. However there is only one hereditarily transitive set of each finite cardinality, and when one exhibits the first few of them the natural reaction is yours: "Oh, he's coding the ordinals as von Neumann ordinals," even though what I was actually doing was exhibiting small hereditarily finite sets.

It turns out that "hereditarily transitive set" is not the most abstract definition of ordinal, in agreement with your intuition. Although you didn't volunteer a formal definition, here's one. Consider all semilattices L with sups less than some regular cardinal &kappa;, equipped with a monotone unary operation s: L &rarr; L. The initial such is the ordinal &kappa;, its elements are the ordinals less than &kappa;, and the monotone operation is successor. The von Neumann ordinals are less abstract in that in effect they make "inflationary" (x &rarr; s(x)) a requirement in addition to "monotone" (x &le; y &rarr; s(x) &le; s(y)). This is unnecessarily concrete because in this larger category without the requirement of "inflationary" the resulting ordinals are inflationary anyway. (Not that I'm proposing to put that in the lead.) --Vaughan Pratt (talk) 22:21, 14 February 2009 (UTC)

Request for help - Possible Error in the article?
In the article is stated that the cardinality of Epsylon-null equals Aleph-null. This is not possible because, Card(Epsylon-null) ≥ Card(Omega^Omega)≥ Card(2^Omega) > Card(Omega) = Aleph-null. I guess I am missing something. Please, help. Boris Spasov, —Preceding unsigned comment added by 161.209.206.1 (talk) 20:53, 22 May 2009 (UTC)


 * What you're missing is that ordinal exponentiation is not the same as cardinal exponentiation. The symbol $$\omega^\omega$$ is (in theory) ambiguous and could mean either of these.  In practice, though, it means ordinal exponentiation, whereas you're apparently interpreting it as cardinal exponentiation. --Trovatore (talk) 05:04, 23 May 2009 (UTC)


 * See Ordinal arithmetic. At the end of the section, it says:
 * Warning: Ordinal exponentiation is quite different from cardinal exponentiation. For example, the ordinal exponentiation 2ω = ω, but the cardinal exponentiation $$2^{\aleph_0}$$ is the cardinality of the continuum which is much larger than $$\aleph_0$$. To avoid confusing ordinal exponentiation with cardinal exponentiation, one can use symbols for ordinals (e.g. ω) in the former and symbols for cardinals (e.g. $$\aleph_0$$) in the latter.
 * The reason for the difference is that cardinal exponentiation counts all functions from the exponent to the base while ordinal exponentiation only counts functions which have finite support (i.e. all but finitely many elements of the exponent are mapped to the zero element of the base). Ordinal exponentiation must be so limited because one cannot effectively define a well-ordering of the power otherwise. JRSpriggs (talk) 09:37, 23 May 2009 (UTC)


 * Yeah, but insofar as this note is about usage rather than mathematics, it needs to be taken with a big grain of salt. It's actually quite common (at least on blackboards) for $$2^\omega$$ to be written for the cardinality of the continuum.  And in fact unless there are clear contextual clues to the contrary you can be almost certain that that is what's intended.  The reason is pretty obvious:  because in the ordinal-exponentiation sense $$2^\omega$$ is just $$\omega\,\!$$, and not for any very deep or interesting reason, there's no reason to write $$2^\omega$$ for it.  Whereas there is a reason to write it for the cardinality of the continuum &mdash; it's just easier to write than $$2^{\aleph_0}$$. --Trovatore (talk) 09:53, 23 May 2009 (UTC)

Thank you for the simple and strait to the point explanation. I consider as a great advantage the very opportunity I got, to ask directly people like you, who know. Regards, Boris Spasov. —Preceding unsigned comment added by 173.51.255.161 (talk) 18:50, 23 May 2009 (UTC)

GA reassessment
Seeing as now the article has virtually no inline citations, and the fact that there are other posts requesting they be added on this talk page, I'm delisting this article. It can be re-nommed when this is fixed. Wizardman 19:33, 6 June 2009 (UTC)

First ordinal of cardinality $$2^{\aleph_0}$$
Is there a standard symbol for the first ordinal of cardinality $$2^{\aleph_0}$$ (equal to $$\omega_1$$ in case the continuum hypothesis holds)? Ninte (talk) 07:51, 19 June 2009 (UTC)
 * No. At least none that's widely understood, and specific to ordinals.  However it's standard to identify the cardinal with the first ordinal of that cardinality, and with that understood you can just call it $$2^{\aleph_0}$$.  The meaning of expressions like $$2^{\aleph_0}+1$$ has to be worked out from context. --Trovatore (talk) 08:52, 19 June 2009 (UTC)


 * The notation $$\mathfrak{c}$$ is also used for the cardiality of the continuum. Again, if you identify cardinals with initial ordinals, you can use this to refer to the corresponding ordinal. &mdash; Carl (CBM · talk) 13:31, 19 June 2009 (UTC)

What do the graphic figures show?
I am not seeing what the figures have to do with the subject of the article -- perhaps an explanation of the connection between the lines etc and ordinal numbers would help.

Specifically:

Are numbers represented by the lengths of the line segments, or the sapcing between them, or both? Why do both get smaller as one follows the spiral inward, or the graph from left to right? Is this intended to show perspective of what's "really" a 3-D graphic? Or maybe I am reading it backwards?

Why are there junctures where the progression jumps from tiny to large again? I'm guessing this is the transition from the last natural number to the start of omega as the base number, for example. But that suggests a huge jump in "size" from last natural number to omega, is that intended?

Bottom line, I guess I'm looking for some mapping between the numbers discussed in the article over to the graphic features in the figures. Gwideman (talk) 00:02, 19 December 2009 (UTC)


 * Perhaps you just misformulated, but if you actually think there is a "last natural number" then you have misunderstood ordinals.
 * Yes, I do understand that N doesn't just end at some point. :-) I should have spelled out: "this is the transition from the last graphic representation of a natural number to the start of the graphic representation of omega as the base..." Gwideman (talk) 01:26, 19 December 2009 (UTC)
 * The large bar at the top of the picture represents 0, the next one (a bit to the right and down) 1, then 2, and so on clockwise. The bar immediately below 0 represents ω. The numbers that approach it from the left must get smaller and smaller because they are infinitely many. The bar below ω is ω&times;2, then ω&times;3 and so on. These bars also must get smaller and smaller because there are infinitely many and the picture is finite. Hans Adler 00:21, 19 December 2009 (UTC)
 * It's true, though, that the height of the bars does not really need to shrink. The question is whether the shrinking heights make the picture clearer and more intuitive, or the opposite.  On balance I think they probably make things clearer, but the question is not trivial. --Trovatore (talk) 00:40, 19 December 2009 (UTC)
 * Necessarily, a graphic that includes infinity, indeed several infinities, is going to need to be somewhat suggestive. My point here is that even the non-infinity-related aspects of the figure rely on guessing what the artist had in mind. It's looking like the line size and spacing is something like e^(-kn), with resetting of the scale for each transition from n to omega, omega to omega^2, etc. That's counter-intuitive in at least three ways, but that's OK if these counter-intuitive mappings are at least pointed out... especially if there's some ultimate payoff, such as the graphic highlighting some aspect of the topic that is otherwise hard to see.  Gwideman (talk) 01:26, 19 December 2009 (UTC)

The caption does say, "Each turn of the spiral represents one power of ω". That seems quite concrete to me. At least this image is more apropos than a big picture of Cantor would be. However, it is not clear to me that the diagram is of &omega;&omega; and not &omega;&middot;&omega;. &mdash; Carl (CBM · talk) 01:31, 19 December 2009 (UTC)


 * Yes I see the caption, but I don't see that in the figure. At one full turn we get to ω1, right? Then what happens at turn 1.5? Aren't we supposed to wait until turn 2 until we get to ω2? If so, what are all the changes of scale between 1.5 and 2?  Or is the caption really saying we get to ωω at 2 full turns? And ωω ω at turn 3? But that also disagrees with the caption.  Gwideman (talk) 09:00, 19 December 2009 (UTC)


 * I see it now, I wasn't looking closely enough. The first turn gets you one copy of &omega;. The second gets you &omega; copies of &omega;, which is &omega;2. The third turn gets you &omega; copies of &omega;2, and so on. &mdash; Carl (CBM · talk) 12:31, 19 December 2009 (UTC)


 * This makes me think about strictly increasing functions from countable ordinals into the real numbers. Perhaps it would be interesting to have a section or article on examples of such functions. What do you think? Unfortunately there is no systematic way to map all countable ordinals into the real numbers while preserving order. JRSpriggs (talk) 17:30, 20 December 2009 (UTC)


 * That does sound interesting, but I don't think it ought to be added unless someone has actually written about it elsewhere. Cheers, — sligocki (talk) 23:34, 20 December 2009 (UTC)

Please make the lead and intro sections more accessible
Hello, I would like to see something along the following lines; what I want is something far, far more accessible to lay readers. The heavier stuff could go in sections after the introduction.
 * Lead: In mathematics, an ordinal number, or just an ordinal, is used to indicate the size of a well-ordered set.
 * Introduction: For finite sets, the corresponding ordinals are just the natural numbers; the fact that a finite set is ordered some particular way does not matter for its size. However, when comparing infinite sets, different orderings make them compare differently even if they have the same cardinality. Ordinals can be added, multiplied, and exponentiated, and these operations obey much of the same laws as ordinary arithmetic (see Ordinal arithmetic).
 * Definitions:
 * .

Yes, I know the style of imprecise statement I am suggesting in the lead flies in the face of the current Wikipedia tradition for mathematical and scientific articles. But I feel the current tradition is fundamentally misguided. The statement about ordering type is quite exact, I see nothing wrong with it (in spite of the objections in the discussion above) except that we are forcing 90% of the readers to read past the lead without understanding a iota. This percentage would have been much higher if not so many potential readers had already decided that the technical WP articles are unreadable and had stopped looking them up.

Either have a fairly arcane but veeery short first statement in the lead, and switch immediately to a more humane language, or just keep the lead a little less precise. Those in need of complete precision should read the definition and later sections anyway. The purpose of the lead is/should be to give as many readers as possible the closest possible idea of what the article is about in as few words as possible. With so many "as possible", some compromises must be struck.

Keep that statement (order types) but provide a little more context, and put it down in the definition section, approximately. Yes, people can look up order type, but then they probably have to look up 20 other terms -- of limited relevance for understanding the fundamental idea of ordinals, but you only know that after looking them up -- until the "head explodes moment" as another reader wrote in another talk page. All right, the actual order type page is among the better ones in this respect, but a good encyclopedia should not require its readers to be almost experts before reading an article. The initial parts of each article should be readable with a vocabulary much more basic than the subject of the article. "Order type" is not much more fundamental or widely known than "Ordinal number".

As to abstract versus concrete definitions, I doubt that the purest essence of the concept can be expressed in any language at all. Every verbal or formulaic definition must work through some form of encoding, and it will always be possible to abstract the "pure" concept further away from any formulation, however abstract it already is. As humans we reach the abstract notions only after working through some practical realizations of the concepts. As such, the purest essence is like knowledge of a city; you only acquire it after walking around for a while. The question of which definition to use is ultimately a practical one; if your purpose is fundamentally pedagogical you should perhaps choose a different definition than the one most practical for some actual computation or theoretical investigation. This being an encyclopedia, the pedagogical aspect should carry some weight. Since this 'pedia is (I believe) quite heavily used by students, it is a good thing to provide just the kind of informations that students need -- after the more introductory sections. (The article is already quite good for students, it's the lead and intro I am worried about.)

Please describe the canonical representation as transitive sets. Most books I have seen, and all my textbooks, said ordinals were those sets. (But see next paragraph). Notice that the WP article Order type says:


 * Every well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes...

I have an introductory text prepared by Skolem in 1957 for his students where he writes (my translation from Norwegian): "In pure set theory it is most suitable to say that a set is ordered, more precisely: ordered by P, when P is a set of ordered pairs ...(details of transitive, total except for non-reflexive relations)... Cantor here introduced the concept of order type (my emphasis). He wrote $$M\simeq N$$ when M can be mapped uniquely (or "clearly"; probably he meant bijectively) to N while keeping the order. It is clear that the relation $$\simeq$$ is reflexive, symmetric and transitive, so that a division into classes arises. The class to which a set belongs is then its order type. (...) Cantor noted how the order types which he called ordinal numbers (my emphasis) of the well-ordered sets could again be ordered according to their sizes, and this ordering is again a well-ordering." Reference: Th. Skolem: Forelesninger i mengdelære (Lectures in set theory), Universitetets Matematiske Institutt, Oslo 1957.

According to Skolem, Cantor had noted that the numbers $$n/(n+1)$$ together with 1 "clearly is of a different order type" (compared to the natural numbers). I find this case particularly instructive; it has a final or greatest member, yet it looks at first sight very similar to the set of natural numbers. This example could be included in the intro, to give a preliminary feeling of the reality of order types. (The order types article has other examples. But the Order Type article also talks about sets that are not well ordered, like the integers and the negative numbers that have no smallest element.

The current introduction (after the lead and the table of contents) has a readable style, but I am afraid that it is not quite exactly correct. The notion of ordinals is not so directly related to the position of a single element. You may perhaps object that the section does not say "of a single element", but that is what comes to my mind when I read it. The text reminds too much of the regular (not set theoretic) ordinal numbers. The notion of set-theoretic ordinal grew out of mappings, a way of comparing the sizes of two sets. We should stick with the concept of size, and explain that order matters for comparisons. We can think of cardinals as trying all possible pairings, and preferring the pairing that makes the sets equal, if such a way can be found. Ordinals works with well-ordered sets, and use the ordering to chose which elements to pair. That is why we fuzz about the ordering having to be a well-ordering: We need it to be able to chose the smallest next element from each set. Non-well-orderings don't have a smallest element, and so we have no preferred way of pairing the sets. Cacadril (talk) 17:09, 4 March 2010 (UTC)
 * It looks like you've put a lot of thought into this, and I haven't yet read your comments with the attention that they deserve. I would make one, perhaps fairly superficial, remark, though:  I don't really agree that the most accessible way to motivate ordinals is via wellordered sets.  I think it's even more basic than that.
 * Put simply, ordinals are the natural continuation of the natural numbers, thought of as "counting numbers", when you count past infinity. That statement as it stands is too simple and too vague even as an introduction; we have to maintain an encyclopedic tone.  But that's the intuition we should be trying to reach.  --Trovatore (talk) 22:22, 4 March 2010 (UTC)
 * I don't think any of us can "count past infinity".   D b f i r s   22:26, 4 March 2010 (UTC)
 * Well, but only in the same sense that you can't count past a trillion; because you don't have time. --Trovatore (talk) 22:27, 4 March 2010 (UTC)
 * I tried, many years ago, when I has measles and was made to stay in bed, but I didn't make even a million!   D b f i r s   22:36, 4 March 2010 (UTC)
 * Apologies for trivialising a serious discussion. I don't have sufficient expertise to contribute usefully, so I shouldn't have commented.    D b f i r s   23:00, 4 March 2010 (UTC)
 * No no, no apologies needed. We're specifically talking about accessibility, so input from non-specialists is helpful.  You seem to be an intermediate case anyway, holding an undergraduate math degree. --Trovatore (talk) 23:13, 4 March 2010 (UTC)
 * Thanks. I didn't include Georg Cantor's work in my degree, so I struggled (even as an Honour's mathematics graduate) with the concept in the introduction. In fact, I had to go to our article on Cardinal number to understand this article.  It is certainly not easily accessible to those who are unfamiliar with set theory.    D b f i r s   07:44, 8 March 2010 (UTC)
 * "In mathematics, an ordinal number, or just an ordinal, is used to indicate the size of a well-ordered set" is dead wrong. Never mind the informal language, the problem is that this does not describe ordinals, sizes of sets are indicated by cardinals. There is no way around referring to order types in one way or another (not necessarily naming them formally as such) in description of ordinals.—Emil J. 11:47, 5 March 2010 (UTC)
 * I agree with EmilJ. The word "size" means cardinality rather than order type, so it is wrong. JRSpriggs (talk) 03:11, 6 March 2010 (UTC)
 * Well, this is not really a fundamental objection. For size put length; no one can interpret that as "cardinality".
 * To me the more serious problem is that talking about "wellordered sets" before you talk about ordinals, then produces an apparent circularity when you go to say what sets are (objects that appear in the stage of the von Neumann hierarchy corresponding to some ordinal). We should strive to give an intuitive picture of ordinals that doesn't rely on sets at all. --Trovatore (talk) 03:20, 6 March 2010 (UTC)
 * Ordinals are a way of extending the natural numbers into the infinite which allows one to continue to do mathematical induction and recursive definitions via transfinite induction. JRSpriggs (talk) 04:23, 6 March 2010 (UTC)

Topological limit?
In section 4.3 (successor and limit ordinals) the article states "that a limit ordinal is indeed the limit in a topological sense". I think its not the limit, but the colimit / direct limit. Just calling it topological limit might be confusing for people with topological background or for someone who tries to look up what this phrase means. Stephanie 85.176.206.253 (talk) 12:49, 16 May 2010 (UTC)


 * We are talking about topology, not category theory. In the order topology, a limit ordinal is a cluster-point (limit) of the set of smaller ordinals. In other words, any open interval {x|&alpha;<x<&beta;} which contains the limit ordinal must contain some smaller ordinals. JRSpriggs (talk) 14:13, 16 May 2010 (UTC)

Some Examples?
It seems to me that this page could be improved with an example or two from the 'well-ordered' page. That is, compare

{ 0, 1, 2, ... } and { 1, 2, .... 0 ) and { 0, 2, 4, 6, ... 1, 3, 5, ... } and their ordinals.

Bmargulies (talk) 15:45, 23 April 2011 (UTC)

Errors?
There seem to be errors in the article, but I don't know if they actually are errors. Could someone with more knowledge check this? For example, in the very first section it says:


 * Two sets S and S'  have the same cardinality if there is a bijection (i.e. one-to-one onto function) f which maps each element x of S to a unique element y = f(x) of S'  and each element y of S'  comes from exactly one such element x of S. S and S'  are order isomorphic if there is a partial ordering < defined on S and a partial ordering <' defined on S' , such that the function f preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Every well ordered set S is order isomorphic to exactly one ordinal number.

That paragraph defines two sets to be "order isomorphic" if a partial order exists that has certain properties. But such a partial order will ALWAYS exist. The "empty" partial order (where nothing in the universe is comparable to anything else) will satisfy this definition. And I don't think the term "order isomorphic" even applies to sets. It applies to posets. So it seems like the correct version of this paragraph would be:


 * Two sets S and S'  have the same cardinality if there is a bijection (i.e. one-to-one onto function) f which maps each element x of S to a unique element y = f(x) of S'  and each element y of S'  comes from exactly one such element x of S. If a partial order < is defined on set S, and a partial order <' is defined on set S' , then the posets (S,<) and (S' ,<') are order isomorphic if there is a function f that preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Every well ordered set (S,<) is order isomorphic to exactly one ordinal number.

I think it's common mathematical slang to call a poset a "set", if it's clear from context that "poset" was meant. Even the term "well ordered set" is guilty of this, because a "well ordered set" isn't a set. It's a (set, order) pair. But for this particular article, I think it would be much clearer for the reader if the article kept the distinction between the two clear.

I won't make the above change myself, but could someone who knows this subject make the appropriate changes? — Preceding unsigned comment added by 70.113.33.136 (talk) 21:16, 3 February 2013 (UTC)


 * Actually, your understanding is fine. I suggest that you go ahead and make the changes. If there are any problems with them, someone will correct you.
 * The only change in what you said that I would suggest is to specify that the ordering on the ordinal is the element relationship. JRSpriggs (talk) 04:11, 4 February 2013 (UTC)

Simplify intro please!
If a non-adept can read the intro and has no idea what's being discussed without looking other terms up, it's a bad intro for an encyclopedia. I'd give you an example, but I don't know what you're talking about. FYI - I have a BS in mechanical Engineering, so I've had my nose rubbed in some math.Pb8bije6a7b6a3w (talk) 15:01, 10 June 2014 (UTC)

Written for mathematicians?
Er, could someone make the first paragraphs of this (and the pages for cardinal and nominal numbers) simpler for laymen? As in, "An ordinal number is a number in a set, i.e. 1st, 2nd, 3rd." - The jargon is confusing, especially the stuff about hereditary transitive sets. 77.99.7.242 (talk) 13:23, 28 April 2010 (UTC)
 * I think you're looking for ordinal number (linguistics). --Trovatore (talk) 01:38, 29 April 2010 (UTC)


 * I came across this term when I was reading about infinity. I am usually not averse to mathematical definitions and jargon, and I understand how countable infinities like Natural numbers and Even numbers have same number of elements. However, I found this article incredibly hard (even after going through multiple links in the introduction for cardinals, well-ordered sets and so on). Is it possible to simplify this topic or is it too technical? Maybe give some concrete example like the table on this page. Also, I would like to take your attention to a well written article from NYTimes hoping it may help to tone down the introduction. Thanks! — Preceding unsigned comment added by 118.95.28.36 (talk) 23:41, 18 August 2012 (UTC)


 * I should have read this before I went to the trouble to write my own complaint, because you've already said it. I agree with you completely, this was written to an expert, not for a layman. I feel like the "genius" whats-his-name on the Big Bang Theory is talking at (not to) me. But then, in my experience, that is the single most consistent mistake made in Wiki. Pb8bije6a7b6a3w (talk) 15:13, 10 June 2014 (UTC)

I agree with the former comment. This is written by mathematicians for mathematicians. I now realises that. Ordinal number (linguistics) is where normal "ordinal numbers" are... But that is not really linguistics? Would it be possible (I wouldn't know how!!) for this article to be named "ordinal numbers in set theory". ...and the other one just "ordinal numbers"? I know the mathematicians will hate that, because you have a very different perspective to me, but I expect most people visiting this page are.... confused! We don't think we are looking for "linguistics" or anything. This is just about the first time I looked something up on Wikipedia and then gave up and looked it up on a desktop encyclopaedia instead! RBJ (talk) 06:20, 10 May 2013 (UTC)


 * I think that most people who reach this article are looking for "ordinal number" in the mathematical sense of order types, not for "first", "second", "third", "fourth", etc.. The linguistic use may be more common, but it is not something that people would normally go to an encyclopedia to study. JRSpriggs (talk) 10:18, 10 May 2013 (UTC)
 * According to the page view stats for the last month, this article gets about 1,000 views per day, whereas Ordinal number (linguistics) about 200. This does not say how many users should be discounted as they end up here in error while looking for the other article, but assuming that most would realize this and follow the hatnote link, the numbers confirm your suspicion. (I wonder what happened on April 12.)—Emil J. 11:03, 10 May 2013 (UTC)

"Finer distinction"
I linked the words "finer distinction" in this article to Comparison of topologies. I hope this is correct: please revert it if wrong. -- The Anome (talk) 13:03, 4 October 2015 (UTC)


 * The difference is not a difference of topologies. The class of cardinals is properly contained in the class of ordinals: there are more ordinals than cardinals, so the difference is not one of topology. The "finer distinction" the article is claiming is that the ordinal number of a well ordering completely characterizes the order type of the well ordering, while the cardinal number does not. &mdash; Carl (CBM · talk) 13:51, 4 October 2015 (UTC)

Maybe, "finer distinction" could be linked to quotient set? I think a cardinal number (viewed as von Neumann ordinal) can be identified with the equivalence class of all ordinals that can be mapped to it by a bijection (i.e. that have the same cardinality), and the least ordinal in that equivalence class is a canonical representative. However, a class, rather than a set, is needed to collect all cardinals, so "quotient class" would be more appropriate. - Jochen Burghardt (talk) 20:36, 4 October 2015 (UTC)
 * I'm usually against this sort of link. "Finer distinction" doesn't in general mean "quotient set", and the piping feature should not be used to find a place for "hey, here's an article you might be interested in reading at this point" by piping some tangentially related bit of text.  If you think it's a useful link, then it would be better to add text that naturally contains the search term you want to link. --Trovatore (talk) 21:01, 4 October 2015 (UTC)


 * I'm sorry to hear the link isn't correct: thanks for removing it. However, if this is not the topological concept, I'd very much like to know exactly what is meant by the words "finer distinction" in this case, and if the same concept is used elsewhere in mathematics. If so, regardless of whether it's a simple idea, or a very difficult one, it probably deserves its own article. -- The Anome (talk) 12:18, 6 October 2015 (UTC)


 * For example "velocity" is a finer concept than "speed" because the latter has all the information of the former plus more information (the velocity has the speed and the direction). Similarly, the ordinal of a well ordering does tell you something about the cardinality of the well ordering (it's the same as the cardinality of the ordinal) but the ordinal tells you more - it tells you the exact order type of the well ordering. There's nothing topological about that. As for "quotient set", if that means "equivalence class" then at least the link is more accurate (because "having the same cardinality" is an equivalence relation on ordinals), but I tend to agree with Trovatore that the link does not add that much value. Normal, nontechnical English uses the phrase "finer distinction" - I don't think we need to search for an article to link to the phrase. But 'quotient set' has a connotation of topology that 'equivalence class' does not, so the latter is better than the former in any case.  &mdash; Carl (CBM · talk) 12:35, 6 October 2015 (UTC)


 * While
 * ... ordinals draw a finer distinction than cardinals on account of their order information
 * is correct, it is a pretty much horrible formulation to have in the lead. It is as made for misunderstanding. What is order information? I know that, but if I didn't know to begin with, I probably wouldn't be able to glean from the context what it is at that point. The posts here call for a reformulation. Something along the lines of between each consecutive pair of cardinal numbers there exists infinitely many ordinal numbers linearly ordered... YohanN7 (talk) 12:52, 6 October 2015 (UTC)

Lede Really Needs to Get Simplified
On this talk page, there have been mentions of the overly technical language within the lede by multiple users, namely (here), (here), and here. This article really needs to be fixed up, and pronto, as the first two of those three talk page sections had been done in 2010. The last of the three had been done over a year ago, in 2014.

I have already added the  template to the article. Come on, people, let's get this done already. – SarahTehCat (talk) 18:59, 1 November 2015 (UTC)

Need a simpler, basic Intro definition of ordinal vs. cardinal numbers
I've been teaching myself set theory over the last few months and Wikipedia has really helped a lot. I just want to acknowledge several of the math articles here, on the difference between the naturals, the reals, etc., the definitions of the ZFC axioms are really well written. But one thing I found lacking was a good definition of the difference between a cardinal and ordinal number. I get it now but when I first came here I found both articles to just dive in deep without first giving the basic idea. IMO a good Wikipedia article should always do that. We need the detail for people like me who are really trying to dig deep into the topic but we also need the basics for someone who just comes across the terms and wants to understand the difference. IMO we need something like "ordinal numbers are for ordering, they tell what position something takes in an ordered group where as cardinal numbers are for counting, they say how many somethings are in a group". I don't feel competent enough to write it myself but just thought I would leave the suggestion here. I'm kind of embarrassed to admit it but I had to go here: https://www.mathsisfun.com/numbers/cardinal-ordinal-nominal.html to first refresh my memory of the basic idea, after that I could get the more detailed discussions that currently exist in Wikipedia. --MadScientistX11 (talk) 15:47, 19 November 2015 (UTC)


 * I would correct your sentence to "Ordinal numbers are for ordering &mdash; they tell what position something takes in a well-ordered set. Whereas cardinal numbers are for counting &mdash; they say how many things are in a set". Using "group" would invite confusion with group theory. There are orderings which are not well-orderings, so we need to be specific. JRSpriggs (talk) 20:37, 19 November 2015 (UTC)

Theorem relevant to the set of countable ordinals
There is a theorem that, in my view, enlightens and even proves many statements about the countable ordinals, such as "the first uncountable ordinal is unreachable by countable operations" (when formalized a bit). The theorem I have in mind is this:
 * Let $Ω$ be the set containing all countable ordinals and $ω_{1}$. Then the least upper bound of every countable subset of $Ω$ not containing $ω_{1}$ is strictly less than $ω_{1}$.

The proof is easy. It becomes obvious that sequences such as
 * ω, ωω, ωω ω, &hellip;

doesn't lead to anything uncountable (given that the terms are countable of course). This proves $ε_{0}$ is countable, not just states it.

I don't know if this theorem should have a place in the article, and if so, how? Opinions welcome. YohanN7 (talk) 12:46, 1 December 2015 (UTC)

In the lede: ordering, total ordering, and well ordering
The lede speaks of "ordering" and, in at least one place, it speaks of "total ordering", but it should instead be speaking of "well ordering". For example, to say that ordinals are about putting "a collection of objects in order, one after another" is false: the ordinal $ω$ is not the immediate successor of an other ordinal. If you have the energy, please fix the lede. 𝕃eegrc (talk) 18:02, 19 February 2016 (UTC)


 * Total ordering is discussed only in connection with finite sets. I have added a mention of well ordering to the first paragraph.  I disagree that the lead is false.  Clearly, interpreting the first sentence to mean "Every ordinal is the successor of some ordinal" is false.  Instead, it should be interpreted as the true statement "Every ordinal has a successor".  Hopefully the new first paragraph addresses these concerns.   S ławomir  Biały  19:35, 19 February 2016 (UTC)

The "order, one after another" part still comes across to me as too easy to misinterpret. First, it could be interpreted as "Every ordinal is the successor of some ordinal", which is false. Second, even if it is interpreted as "Every ordinal has a successor", this property is not defining. It is not the case that all ordered sets with this property are well ordered; consider the set of all integers. Unless someone else beats me to it, I will try again to find a wording that achieves consensus. 𝕃eegrc (talk) 13:07, 23 February 2016 (UTC)


 * There is a danger at pitching the first sentence at too high a level. Ordinal numbers for "most" readers really are just the counting numbers that we use to place numbers one after the other.  So any proposed wording should be understandable by (say) high school children, because ordinal numbers are already used at that level.  Nuances to do with what we mean do not have to be expressed immediately in the first sentence.  There is a whole paragraph (and indeed, an entire article) in which to do that.   S ławomir  Biały  13:29, 23 February 2016 (UTC)


 * It looks pretty good now.  S ławomir  Biały  15:24, 23 February 2016 (UTC)

do you people have any idea
of what a *general* encyclopedia is ?/ this article is way way way to complex, at least in the intro throw it out and start over; if you ten year old doesn't get it instantly, it is to complex sheesh — Preceding unsigned comment added by 2601:192:4200:1E42:4C36:29D7:2036:EFF5 (talk) 16:38, 28 February 2016 (UTC)
 * I worry that throwing it out and starting all over, won't address most of your concerns. Would you be more specific in what you would like to see changed?  𝕃eegrc (talk) 13:51, 29 February 2016 (UTC)

WP is not a textbook
Please see WP:NOT, and please stop writing in ways that address the reader directly as if they're a high school student reading a textbook. It's the wrong tone. (For guidelines as well as the policy, see MOS:YOU, WP:SELFREF, and WP:TONE.) Many of our maths articles have this problem. The two most obvious faults of this sort are a) telling the reader what to do ("Note that...", "It should be observed that..."), and b) asking the reader questions ("What is the meaning of ...?" "How do we know that...?"). Just don't do it. There is never, ever any reason to do either of these sorts of things in encyclopedic writing. — SMcCandlish ☺ ☏ ¢ ≽ʌⱷ҅ᴥⱷʌ≼  22:11, 13 March 2016 (UTC)


 * Please see MOS:MATH. While that section mostly agrees with you, there are a range of opinions about conversational style among math editors. In particular, a little conversational style can help in advanced math articles. The talk page has some discussion of this. --Mark viking (talk) 22:56, 13 March 2016 (UTC)
 * I find myself in rare agreement with Stanton here. The article is too chatty overall.  We don't necessarily have to be utterly rigid about it, but though one of Wikipedia's great strengths is as a tool for self-teaching, it is not supposed to actively teach.
 * There is a tension between that goal and the one, especially important to me in this article, of getting across the informal intuition. I think it took me quite a while into my graduate studies before I fully internalized that intuition myself, so it may be unrealistic to think we can do it in an article meant to be read in half an hour, but I would like to try.  I think we should try to address that goal within the bounds of encyclopedic tone. --Trovatore (talk) 23:08, 13 March 2016 (UTC)
 * Assuming this discussion concerns these edits, I fully agree. But sometimes it actually is appropriate and correct in strict, formal mathematical prose to address the reader (e.g., "Suppose that a set X is given...").  There is a spectrum between inane textbookish phrases such as those just removed, and the kind of addressing the reader that is actually necessary for writing formal mathematics in a way that can be read by other mathematicians.  For example, the use of the first person plural pronoun "we" is widely accepted in formal mathematical discourse ("Suppose that we are given a set X...).  One can find this pronoun used in research papers of the highest quality, mathematics encyclopedias, research monographs, and textbooks of all levels.  My own general attitude is that such use should be minimized, but if it turns out to be the least awkward way to express a complicated idea, then it would be a mistake to do it any other way.  This is consistent with the demands of WP:TONE, which enjoins us to "follow the style used by reliable sources."   S ławomir  Biały  11:45, 14 March 2016 (UTC)

Intro too specific to von Neumann ordinals
The third and fourth sentences of the opening paragraph are:
 * The ordinals extend this concept beyond the natural numbers by having each ordinal represent the set of all ordinals that are less than it. For instance, the ordinal 2 represents the set $\{0, 1\}$ and the smallest infinite ordinal, written $ω$, represents the set of all natural numbers, $\{0, 1, 2, …\}$.

It seems to me that this is too specific to the von Neumann representation of ordinals. I would prefer to try to get across the notion that we might call an "abstract ordinal", the one Cantor used, before we get into how to code them up as sets. --Trovatore (talk) 19:42, 29 February 2016 (UTC)
 * The lede doesn't have to give the definition based upon the earliest historical formulation, but it is nice when it can. Edit at will!  𝕃eegrc (talk) 21:14, 29 February 2016 (UTC)
 * I made an edit. Instead of going with Cantor, I indicated that this formulation in the lede is one of many.  Does that help?  𝕃eegrc (talk) 21:20, 29 February 2016 (UTC)
 * Well, it's a little better. I would still prefer to get the underlying intuition across first, before talking about representations at all.  I might not have explained myself perfectly; I'm not going for historical fidelity here, particularly.  Bringing in Cantor was just a way to make people remember, oh right, the concept can be described without reference to a representation; it must be possible, because Cantor did it. --Trovatore (talk) 21:24, 29 February 2016 (UTC)
 * The lead is an introduction, at least in this article, so I am OK with putting an informal description of the standard concrete representation up front. However we could add something like,
 * ...For instance, the ordinal 2 represents the set $\{0, 1\}$ and the smallest infinite ordinal, written $ω$, represents the set of all finite, natural numbers, $\{0, 1, 2, …\}$. More abstractly, ordinal numbers can be thought of as labels for equivalence classes of well ordered sets over one-to-one mappings that preserve order.
 * --Mark viking (talk)
 * That's not the abstraction I meant. That's not intuitive enough.  I meant, we should start with the idea of $0, 1, …, ω, ω+1, …, ω⋅2, ω⋅2+1,…$.  We should get the informal intuition across before we start talking about concrete representations of any kind. --Trovatore (talk) 21:55, 29 February 2016 (UTC)

I tried something like that here, but "one after another" rubbed some editors the wrong way.  S ławomir Biały  23:46, 29 February 2016 (UTC)
 * Yeah, I liked that wording myself, but I could see the objectors' point; it's not clear that a naive reader will interpret it correctly. Can we come up with some wording that addresses that objection? --Trovatore (talk) 07:26, 1 March 2016 (UTC)
 * Is the argument that the "$0, 1, …, ω, ω+1, …, ω⋅2, ω⋅2+1,…$" representation is more abstract than the von Neumann representation? 𝕃eegrc (talk) 12:47, 1 March 2016 (UTC)
 * Hmm? It's not a representation at all.  I'm not talking about the symbols.  I'm talking about the underlying intuition. --Trovatore (talk) 17:34, 1 March 2016 (UTC)
 * Regardless of whether we call it a representation, I think you are saying that writing the likes of "$0, 1, …, ω, ω+1, …, ω⋅2, ω⋅2+1,…$" is not more or less abstract than writing the likes of "$\{0, 1\}$" and "$\{0, 1, 2, …\}$"? But, I am confused as to what you do mean.  Can you give a snippet of what you might write that does achieve abstractness?  𝕃eegrc (talk) 21:06, 1 March 2016 (UTC)
 * I think the intuitive idea was conveyed by the version of the lead that you rewrote. The ordinals are abstract labels used to put sets "in order".  Identifying what one means by "in order" is a little bit tricky.  For finite sets, this means putting them "one after another".  For infinite sets, it still means putting them one after another (successor ordinals), but we're also allowed to adjoin labels that follow infinite collections (limit ordinals).   S ławomir  Biały  21:18, 1 March 2016 (UTC)
 * I've edited again. Even if it doesn't get all the way to what you are hoping for, I am hopeful that it is in the right direction.  I believe there is general consensus that an understandable abstract definition of an ordinal is the ideal; all we have to do is come up with the wording.  Please give it a try.  𝕃eegrc (talk) 15:23, 2 March 2016 (UTC)


 * I thank you for your bold edit Slawekb, even though it undoes my attempts. I am still hoping for a version that is not only clear but also clearly true.  I leave it to the rest of you to give it a shot.  𝕃eegrc (talk) 16:41, 2 March 2016 (UTC)

The ordinals should be presented axiomatically together with categoricity of the axiomatization.

See Inconsistency Robustness in Foundations: Mathematics self proves its own formal consistency and other matters

Carl (talk) 15:00, 22 October 2016 (UTC)


 * That seems inappropriate, as being only correct in your logic, and being no more intuitive than abstract order types. However, if we were to work together with drafts on this talk page, we should be able to come up with a lead which is both correct and intuitive.  — Arthur Rubin  (talk) 04:44, 23 October 2016 (UTC)


 * Actually, the categoricity theorem for ordinals in the above referenced article would have been acceptable to Dedekind. The axiomatic approach has important advantages over the current presentation.


 * Carl (talk) 22:32, 24 October 2016 (UTC)


 * Propose specific text with a classical source. A (second-order) axiomatization of On could have a place in the article, and, if simple enough, could be used in the lead. However, it probably should not replace the current approach.
 * "Inconsistency Robustness", although "reliable" in a sense, is not clearly trying to discuss the same topic, so it would need to have separate articles in Wikipedia, if it were to be included, at all. (BTW, I found what the error in your analysis would be in classical logic, but it's inappropriate here.  With your permission, I'll post on your home page, later.)  — Arthur Rubin  (talk) 17:26, 25 October 2016 (UTC)
 * The above-referenced article is not about inconsistency robust logic. Instead, it is entirely about the community of classical mathematicians. Carl (talk) 18:16, 25 October 2016 (UTC)
 * Axiomatization of the ordinals in the above-referenced article is based on strong parameterized types used in computer science. Carl (talk) 18:21, 25 October 2016 (UTC)
 * That's your opinion. Doesn't seem justified, from what I've seen of the article. — Arthur Rubin  (talk)
 * That the ordinals are categorically axiomatized by the axioms is dominant in computer science.
 * [User:Prof. Carl Hewitt|Carl]] (talk) 00:57, 26 October 2016 (UTC)
 * That is also not relevant, absent evidence that abstract ordinal number types are actually used in (theoretical) computer science. — Arthur Rubin  (talk)
 * Computer Science needs strong categorical axioms for all of classical mathematics so that computers can participate in proving all mathematical theorems without being be hacked by inconsistencies.
 * Carl (talk) 00:57, 26 October 2016 (UTC)
 * — Arthur Rubin (talk) 18:49, 25 October 2016 (UTC)

Sacks
Who is Sacks and what text is referred to in the article? 145.118.72.144 (talk) 17:07, 27 March 2010 (UTC)
 * Sacks is clearly Gerald Sacks. Sacks' book Mathematical logic in the 20th century, was reprinted in 2003, but it doesn't seem to make much sense to cite it in this way. None of the older papers it reprints gives an in-depth introduction to ordinal numbers, and Sacks' introduction to the book doesn't go into any detail. Hans Adler 17:22, 27 March 2010 (UTC)


 * , it would only be "clearly" Sacks to non-lay users. – SarahTehCat (talk) 17:32, 1 November 2015 (UTC)


 * That is why I, as a non-lay user who had looked at the article and its sources in detail at the time (actually much more important than the 'non-lay' part) explained what I think it is supposed to mean. It was a contribution towards fixing the problem, not an attempt to defend it. Hans Adler 10:09, 4 July 2019 (UTC)


 * I am sure that is supposed to be Jech 2003, looking at the references section below. &mdash; Carl (CBM · talk) 14:14, 28 April 2010 (UTC)

Elaborate graphic representation
Could sb please clarify a bit the visual representation "Each turn of the spiral represents one power of ω"? --Backinstadiums (talk) 14:57, 11 July 2019 (UTC)
 * It seems to me that this is true, but it isn't actually needed. A slightly more precise, but far more clumsy, version would be "The outermost n turns of the spiral represent the ordinal numbers up to ωn". - Jochen Burghardt (talk) 16:10, 11 July 2019 (UTC)


 * A strictly increasing function from &omega;1 into the real numbers is impossible. So any pictorial representation of the ordinals is bound to have some limitations/problems. JRSpriggs (talk) 00:49, 12 July 2019 (UTC)
 * I see what you're saying, but I can't say I see it as a big problem. The graphic is presented as representing the ordinals up to &omega;&omega;, not all the ordinals.  I think it gives a decent intuition of how the ordinals proceed even after that.  To be sure, you can't embed them all into a 1-D submanifold of the real plane, but qualitatively they continue in the same way, being built up as successors and limits of what came before. --Trovatore (talk) 04:16, 12 July 2019 (UTC)

Error: The $$\omega^3 + 2$$ on the bottom of the third spiral in the title image should say $$\omega^2 + 2$$
EDIT: I misread the diagram; it's correct. Nonetheless, a bit more spacing between the inner spirals would potentially be helpful. — Preceding unsigned comment added by Botosuhna (talk • contribs) 06:02, 26 August 2019 (UTC)

See previous revisions of the image. — Preceding unsigned comment added by Botosuhna (talk • contribs) 05:54, 26 August 2019 (UTC)

Intro for a non-technical audience
Can we find a way of the intro explaining the key principle in simple terms for a non-technical audience. Wikipedia is for everyone. If someone comes across the term 'ordinal number' they should be able to reference Wikipedia to get a explanation, that they understand 86.11.51.106 (talk) 06:52, 28 September 2020 (UTC)
 * I agree that the first sentence is hard to understand for non-mathematical readers. But the modified lede was wrong, so I reverted it. It stated "... an ordinal number ... is a natural number ..." That's wrong. All natural numbers are ordinal numbers, but the reverse is not true: Almost all ordinal numbers are not natural numbers. Only the finite ordinal numbers are natural numbers, but almost all ordinal numbers are infinite. (That's what makes ordinal numbers interesting and relevant.) -- Chrisahn (talk) 09:58, 28 September 2020 (UTC)


 * I think the paragraph starting with "Perhaps a clearer intuition of ordinals..." is a reasonably good introduction to infinite ordinal numbers for people who haven't heard of the concept before. Maybe we should move this paragraph into a section of its own, and maybe this section should come right after the lede. Maybe that would be a good compromise: The lede would be short and correct, the next section would be helpful for lay people. (What should be the name of that section? "Non-technical introduction"? "Intuition"? ...?) -- Chrisahn (talk) 10:16, 28 September 2020 (UTC)
 * I think is probably expecting the concept treated at ordinal numeral.  That's not actually an unreasonable expectation.  We have a hatnote about it but it's easy to miss. --Trovatore (talk) 16:43, 28 September 2020 (UTC)
 * I think is probably expecting the concept treated at ordinal numeral.  That's not actually an unreasonable expectation.  We have a hatnote about it but it's easy to miss. --Trovatore (talk) 16:43, 28 September 2020 (UTC)