# Talk:Transfinite number

WikiProject Mathematics (Rated Start-class, High-priority)

## presentation method

The method of presenting transfinite numbers looks deficient, because the symbols used are not calculating by virtue of their visual characteristics ,but the calculations are made in an extra-numeral manner, for example: what is the result of ω+ω or aleph null + aleph null ,there is nothing in ω or aleph null as symboles to tell us the result . I do think that a better method of symbolizing these transfinite numbers is needed,symboles that should calculate by virtue of their own visual characteristics.

What sorts of things are transfinite numbers used for? I would find it useful if the article mentioned that.

Hm, if you write '2+3=5' are you then 'calculating by the visual characteristic of the numbers?' I think not. If you wrote 'II + III = IIIII' then yes, but if you add 2 and 3 there's nothing the symbols would tell you about result being 5 (you have to go back to the definition of the symbols and the Peano axioms to get this result). And exactly the same holds here. If you want to add 'omega+omega' you have to look at the definition of omega and the axioms that hold for addition of ordinal numbers. 134.169.77.186 (talk) 15:43, 21 January 2009 (UTC) (ezander)

## merge?

Shouldn't this be merged with infinate numbers?

Not really, because the phrase 'infinite number' is a naive term, with numerous undesirable connotations, dissimilar to the phrase 'transfinite number'--a formal concept of set theory.-Daniel, [[CSUCI]] 20:30, 10 October 2006 (UTC)

## origins in Europe?

I don't know who was the first to consider infinite numbers in Europe, but it certainly wasn't Cantor. Indeed, Newton himself did some treatment of sequences. The modern formal understanding of infinity begins with theorems by Cauchy, Riemann, Weierstrauss, and others, in the early 19th century. It is certainly true that Indian mathematicians were the first to look at infinite processes. In particular, they were probably the first to define the derivative. Nonetheless, a true understanding of these matters didn't come until much later, with the rigourization of mathematics in the 19th century. Grokmoo 16:26, 15 December 2005 (UTC)

Cantor was certainly not the first person to consider a completed infinite, but he was the one who coined the term "transfinite". And really it's the term that should be the focus of this article, because everything else is just duplication of subject matter covered in other articles.
The bit about the Jaina I find quite unconvincing. The web link says they had the notion that there were different sizes of infinity; well, good for them, but unless they had examples and arguments similar to those of set theory, then it's really not the same thing, or at least there's no good evidence that it's the same thing. --Trovatore 21:59, 1 March 2006 (UTC)

## Jain concept

In response to the message Trovatore sent me:

I'm not sure what you mean about the Jaina concepts not being transfinite numbers. A transfinite number is defined as an infinite number, or a number that is not finite. The Jains may not have developed a system of transfinite numbers nearly as general or advanced as the modern theory, but the fact remains that they did at least develop a system of cardinal transfinite numbers, and more importantly, the basic idea of a transfinite number. I have given some more references that also refer to their system of infinite numbers as "transfinite numbers". Jagged 85 18:13, 22 March 2006 (UTC)

Jag, I'm sorry, but you're simply mistaken; that is not at all the definition of "transfinite number". There are lots of notions that fit that definition; say, the +∞ and -∞ of the extended real number line, the ∞ of the Riemann sphere, infinite numbers used in nonstandard analysis, or infinite surreal numbers. None of these are called "transfinite". The term "transfinite" is reserved specifically for infinite cardinal numbers and ordinal numbers of the sort described by Cantor. Without evidence that the Jaina were talking specifically about these, your edits are inappropriate. --Trovatore 21:34, 22 March 2006 (UTC)
I must agree; Cantor discovered, among much else, that a set infinite in two dimensions was not greater than a set infinite in one dimension. The Jains must therefore be talking about a different, and more intuitive concept (most likely one which cannot be made rigorous), not the modern transfinites. Septentrionalis 22:29, 22 March 2006 (UTC)

Trovatore, if the definition of a transfinite number is an "infinite cardinal or ordinal number" as you say, then the Jains did indeed develop the concept of a cardinal transfinite. In my previous edit, I have already wrote that the Jains discovered the aleph-null, the smallest cardinal infinite number, which is unquestionably a transfinite number. You can confirm this in one of the several peer-reviewed publications I've given as sources. On the other hand, you have not provided any sources that prove otherwise. On what grounds are your views more reliable than the words of peer-reviewed publications written by experts in the field who have actually studied these Jaina texts? Jagged 85 03:15, 23 March 2006 (UTC)

OK, let's look at your refs. First, the Singh reference I cannot find, either on Amazon, or at the York University or University of Toronto libraries. If you have an ISBN there would at least be some place to start.
The L. C. Jain references I haven't found either (haven't looked as hard).
Now to the ones available on the web: The MacTutor articles simply do not support your claims; the word "transfinite" does not appear and the explanations are all vague. The Agrawal reference directly claims many of the things you claim, but the evidence offered doesn't make sense, says nothing about one-one matchings, and says things that appear to directly contradict what we know about the Cantorian concepts (for example, the "infinity in one and two directions" stuff). Moreover its hosting on the infinityfoundation website is frankly not a good sign (take a look at the home page of the website; does not exactly look like a reputable source for mathematics).
I strongly suspect all of your "pro" sources have an agenda, and I think you do, too. Now some of that agenda may have some justice to it; it may well be the case that mathematics from Asia has been undervalued in Occidental history. In fact, given human nature, it's almost certain. But from those of your sources that I've been able to find, it's stretching a point beyond recognition to claim that these Jain concepts are the same thing we now call "transfinite". --Trovatore 04:02, 23 March 2006 (UTC)
I moved all of the text over to the article on infinity, which already has an extensive review of historical concepts of infinity. Trying to shoe-horm this bit of ancient Hindi math into this article seems inappropriate. (Note: I simply moved the text; I pass no judgment on its correctness or accuracy or suitability in general for WP). The rest of this conversation should probaly move to the talk page there as well. 01:10, 25 March 2006 (UTC)
OK, so the problem with putting it at infinity is that most of the content was already there, in the paragraph just above where you moved the material from this article. The other problem is that Jagged's references do (judging from their titles; I haven't actually found them, except for the one extremely dubious Web reference) use the word "transfinite" for the Jain concept. That may be enough to justify, in this article, a reference to the claim that the Jain had the notion of transfinite number. (For the reasons I've already given, that claim should not be asserted as fact.)
It would be good if we could actually see these references. Jagged refers to them as "peer reviewed", but only one of them was in a journal, and it was a history journal. Therefore the peer review was presumably by historians. Nothing against historians as a class (my sister actually married one) but I don't think most of them are qualified to judge this issue. --Trovatore 01:28, 25 March 2006 (UTC)
Look for The Crest of the Peacock: Non-European Roots of Mathematics, which has certainly been peer reviewed and should be available in most libraries (and is by no means "pro" in any way). The section on Jaina mathematics certainly refers to their system of infinite numbers as "transfinite numbers". Jagged 85 22:50, 25 March 2006 (UTC)
It appears that York has that one. Maybe I'll check it out when it's convenient. However I have my doubts about your claim that it's not "pro" in any way—the title strongly suggests it's a work of advocacy (again, that doesn't mean in itself that it's wrong, just that I doubt it's neutral). And the fact that this author uses the word doesn't mean much unless the author is a mathematician. --Trovatore 23:04, 25 March 2006 (UTC)
All that said, I'm reasonably happy with the section as it currently stands. --Trovatore 23:07, 25 March 2006 (UTC)

I noticed at Infinite#Early_Indian_views_of_infinity that the quote

The earliest known documented knowledge of infinity was presented in ancient India in the Yajur Veda (c. 1200900 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

is equivalent to one of the definitions that Suppes and Rubin give for the definition of transfinite, namely:

• Patrick Suppes Axiomatic Set Theory Dover, 1972, ISBN 0486616304, Theorem 58, page 155:
${\displaystyle \mathbf {m} }$ is a transfinite cardinal if and only if
${\displaystyle \mathbf {m} +1=\mathbf {m} }$

Jagged's claim about transfinite -- as now defined to mean Dedekind infinite -- being Indian in origin seems well grounded to me. --Michael C. Price talk 00:57, 30 July 2006 (UTC)

## Definition of transfinite

According to a book I'm reading (Suppes, Axiomatic Set Theory) a transfinite can be defined by the property:

${\displaystyle T=T+1}$ (theorem 58, section 5.3, page 155 (Dover ed))

whereas (loosely) speaking a Dedekind infinite cardinal, ${\displaystyle I}$, is defined by

${\displaystyle I=nI}$

where n is any finite number.

Not all transfinites are infinite, but these two classes become equivalent when we assume the axiom of choice. If true, would this not be worth mentioning in the article? --Michael C. Price talk 12:51, 23 July 2006 (UTC)

I'm sorry, but I can't make sense of what you write above. "Not all transfinites are infinite"? Are you saying that there's a set that has a bijection with a proper extension of it, but is not Dedekind infinite? That's easily false; nothing to do with the axiom of choice. (What's possible without AC is that there's a set that's infinite, in the sense that it's cardinality is not a natural number, and yet the set is Dedekind finite.)
My mistake, I got that the wrong way around. All transfinite cardinals are infinite cardinals. With the adoption of AC then transfinite and infinite become equivalents. I'll strike out and correct my erroneous statements below; see if they still make sense: --Michael C. Price talk 09:44, 25 July 2006 (UTC)
In any case I don't think this article is the right place for such an observation, assuming you can clean it up to something meaningful and true. This is not the distinction between "transfinite" and "infinite". That's a historical distinction introduced by Cantor, to avoid using the word "infinite" in contexts where most of us have no problem using it. --Trovatore 00:52, 24 July 2006 (UTC)
Re "Not all transfinites are infiniteinfinites are transfinite". Suppes page 155 says "It should be clear from earlier remarks that every known proof that a cardinal is transfinite if and only if it is infinite depends on the axiom of choice". --Michael C. Price talk 06:19, 24 July 2006 (UTC)

A nice illustration of the difference between transfinite and infinite cardinals is given by Suppes on page 156, theorem 62:

If m and n are transfinite cardinals then
${\displaystyle m+n\leq mn}$

whereas if they were infinite cardinals we assume AC we have an equality. --Michael C. Price talk 09:25, 24 July 2006 (UTC)

I'm pretty sure you've got the definition of infinite wrong (or at least, non-standard).
M (or the cardinal m) is transfinite if (any of the equivalent expressions below hold)
${\displaystyle \aleph _{0}\leq \mathbf {m} }$
${\displaystyle \mathbf {m} +1=\mathbf {m} }$
There is an injection from M to M which is not a surjection
There is an injection from ω to M
M (or the cardinal m) is infinite if
m is not equal to any natural number (as a cardinal).
There is no map from any natural number (as a set) into (or onto) M.
Shall we go on from there? — Arthur Rubin | (talk) 14:19, 24 July 2006 (UTC)
Yes, my mistake, sorry. --Michael C. Price talk 09:44, 25 July 2006 (UTC)
Ah what? You seem to be using "transfinite" as a synonym for "Dedekind infinite". I am not familiar with that usage; I use "transfinite" and "infinite" synonymously (in the context of sets, that is; of course it doesn't make sense to call the extended real number +∞ "transfinite"). As I understand it, the reason for introducing the term "transfinite" was to avoid implying that these objects were absolutely infinite, in a philosophical or theological context. --Trovatore 16:44, 24 July 2006 (UTC)
Arthur's and Suppes' definition are the same: "Dedekind infinite" and "transfinite" are synonymous in cardinality terms. --Michael C. Price talk 09:44, 25 July 2006 (UTC)

Thanks for the clarifications. I should state (although I'm sure it's pretty obvious) that I'm not mathematican; I'm just trying to relate what I'm reading in Suppes with what I see on Wikipedia (as I'm sure many others are also). Your 2nd definition of transfinite is the same as the one I gave from Suppes, so I'm happy with that. I'm not so sure about infinites -- and I did say I was speaking loosely when I proferred my "definition" -- but they look okay. Moving from definitions to properties, do you agree with Suppes' theorem (on page 156, theorem 62, Dover ed) I gave above, namely that:

If m and n are transfinite cardinals then
${\displaystyle m+n\leq mn}$

Do you have Suppes' "Axiomatic Set Theory", ISBN 0486616304, to hand (i.e. can I just quote page and theorem numbers)? --Michael C. Price talk 15:04, 24 July 2006 (UTC)

I don't have Suppes' book, but I concur with that theorem, with the one-line proof
${\displaystyle m+n\leq 1+m+n+mn=(1+m)(1+n)=mn}$Arthur Rubin | (talk) 16:49, 24 July 2006 (UTC)

Yup, that's how Suppes does it as well (great minds think alike, obviously). I presume you agree that these become equalities when we are dealing with infinite cardinals (= alephs?) assume AC? If so this seems worth stating in the article. --Michael C. Price talk 17:01, 24 July 2006 (UTC)

If choice fails, then infinite cardinals do not have to be alephs. I think you're still thinking there's a distinction between "transfinite" and "infinite" that is not in fact standard usage (perhaps Suppes uses such a distinction; if so, it's ahistorical and unfortunate). --Trovatore 17:04, 24 July 2006 (UTC)

You're right am still thinking that. I thought that if choice fails that all transinfinites do not necessarily have to be intransfinites. Even if Suppes' distinction is ahistorical and non-standard then it still deserves to be documented, even if only to avoid confusion. I went through a similar process over at axiom of choice about how Suppes defined "axiom of choice": initially I was told Suppes' definition was wrong, then that it was non-standard and (eventually) that it was equivalent...! (The confusion arose over how "choice" was defined -- there are various formulations it seems.) --Michael C. Price talk 17:12, 24 July 2006 (UTC)

Standard terminology, in a context where we don't know AC, is as follows: infinite means not equinumerous with any natural number, Dedekind infinite means not equinumerous with any some proper subset of itself. Transfinite, as far as I know, is simply a synonym of "infinite", except that it is used only in the context of ordinals and cardinals, and not of other sorts of infinite quantities (such as extended reals or hyperreals or surreal numbers). An aleph is a wellordered cardinal, as opposed to, say, the cardinality of the continuum, which (if choice fails) may not be equinumerous with any ordinal at all. --Trovatore 17:18, 24 July 2006 (UTC)

Certainly Suppes uses the same definition of Dedekind infinite as Trovatore has given. Have to check up about the others, but he certainly says that

${\displaystyle \mathbf {m} +1=\mathbf {m} }$

is an equivalent way of defining transfinite cardinals. --Michael C. Price talk 17:53, 24 July 2006 (UTC)

In set theory, transfinite is often used as synonymous to "Dedekind infinite" (with the definition ${\displaystyle \aleph _{0}\leq \mathbf {m} }$). In mathematical logic, I've seen it used as a synonym of "infinite" (in the context of ordinals or cardinals), but I'm not sure what the most common definition is. This may lead to problems with this article, I suppose.... — Arthur Rubin | (talk) 17:48, 24 July 2006 (UTC)
OK, I'm with you now. --Michael C. Price talk 09:44, 25 July 2006 (UTC)

Darn, that complicates things. --Michael C. Price talk 17:53, 24 July 2006 (UTC)

I'm a set theorist and am not familiar with the usage Arthur claims is common in set theory. Of course I rarely work with models so pathological that "infinite" and "Dedekind infinite" are not the same thing (choice has to fail quite badly for that to happen). But in any case it seems ahistorical; as I understand it, Cantor introduced the term "transfinite" to describe objects that were beyond a limit (trans-finite), but not necessarily without limit (in-finite). Clearly that has nothing to do with the infinite/Dedekind infinite distinction. --Trovatore 18:07, 24 July 2006 (UTC)
Yes, your (and Arthur's) understanding of the distinction makes sense to me now. Thanks! There's a lot of stuff here that can go in the article. I suggest we include the inequalities above and mention AC more. --Michael C. Price talk 09:44, 25 July 2006 (UTC)
I take it you mean two separate things by "my understanding" and "Arthur's", since he and I don't seem to share a common one. Frankly I'm not excited about expanding this article much; it does seem that we'll have to document Suppes' usage, which I think is a minority one (maybe we should poll other logicians about this, say JRSpriggs and CMummert and whatever other name I'm not coming up with at the moment). But since I don't believe that usage is terribly standard, I wouldn't really like to see a lot of substantive information on it here. It would go more naturally at Dedekind finite set. --Trovatore 15:24, 25 July 2006 (UTC)
No, I think Arthur, Suppes and you are all saying almost the same thing. I should have said "same understanding". The only point on which you and Arthur/Suppes might disagree is where you say:
Clearly that has nothing to do with the infinite/Dedekind infinite distinction.
since, AFAICS, "transfinite" is virtually synonymous with "Dedekind infinite", which in turn becomes synonymous with "infinite" when we adopt the axiom of choice. --Michael C. Price talk 18:27, 25 July 2006 (UTC)
I don't seem to have made myself clear. My understanding of the meaning of "transfinite" has nothing special to do with Dedekind infinite. As I understand it, the term was introduced to distinguish objects like ${\displaystyle \aleph _{17}}$, which are transfinite, from absolutely infinite collections—what we would now call proper classes. Since it is now standard to refer to ${\displaystyle \aleph _{17}}$ as "infinite", and to say "proper class" when one means "proper class", the term is no longer really needed, but it survives on a sort of linguistic inertia.
Suppes' use of the term as meaning distinctively "Dedekind infinite", on the other hand, is not inherently bad, but it's not very well motivated, and I don't think it's widespread enough that the information about Dedekind infinite sets should go under this title. What we should do is add a brief notice that this usage exists, and point the reader to Dedekind-infinite set for more information on the topic. --Trovatore 19:24, 25 July 2006 (UTC)
I find it hard to believe that Suppes' usage is that non-standard; what I can believe is that the distinction is now irrelevant to most mathematicans since the axiom of choice is now widely adopted. But in either case, I agree, the usage should be documented. --Michael C. Price talk 21:07, 25 July 2006 (UTC)
I repeat, I think Suppes' usage is non-standard; it may be that a fair number of mathematicians know it, but you can't count on being understood if you use it, whereas if you say "Dedekind infinite", you can. The historical motivation for using the word "transfinite" rather than "infinite" was utterly unrelated to the Suppesian distinction (recall that the word was coined by Cantor, at a time when the axiom of choice had not even been formulated). Yes, the usage should be mentioned here—but any deep analysis of the notion should go at Dedekind-infinite set. --Trovatore 21:20, 25 July 2006 (UTC)
It sounds as if Suppes' use is obscure, rather than non-standard and that the obscurity is due to the widespread acceptance of AC. You are undoubtedly correct that using the phrase "Dedekind infinite" is more likely to be understood than "transfinite", otherwise we wouldn't be having this conversation. But being obscure, IMO, makes it more important to document it, not less. Otherwise why not just redirect "transfinite" to "infinite"? Personally I found your vernacular description Cantor introduced the term "transfinite" to describe objects that were beyond a limit (trans-finite), but not necessarily without limit (in-finite) very helpful; this would be good addition to the article. (I didn't quite see the relevance of the axiom of choice post-dating Cantor; the relevant point is that the notion of Dedekind-infinite pre-dated Cantor, isn't it?) --Michael C. Price talk 21:50, 25 July 2006 (UTC)
But don't you see that my "vernacular description" relates to the "infinite-but-not-necessarily-absolutely-infinte" notion, rather than the "Dedekind infinite" one? Something that's infinite, in the classical sense of the word, is without any limit at all. It cannot be gathered into a completed whole. In modern terminology, it's a proper class (if it's mathematical at all, that is).
The alephs, on the other hand, can be considered as completed wholes. So they're not classically infinite. But neither are they "finite" in the ordinary sense (for example, because they're larger than every natural number). They exceed a limit—a limit point of the ordinals, for example (though you could quibble about ω itself here—it's the first limit, not strictly bigger than one). But they're not without limit, the way that the (proper) class of all ordinals is.
This has nothing to do with AC (it can all be phrased in terms of ordinals, which are automatically wellordered, and where automatically the notions of "infinite" and "Dedekind infinite" are the same). Rather, it has more to do with limitation of size. --Trovatore 22:11, 25 July 2006 (UTC)
I really do take on board the notion that "infinite" is in some sense a less constrained and more inclusive concept than "transfinite", but isn't this encapsulated in the Suppesian statement that "all transfinites are infinite" (which is independent of AC) whereas the converse "all infinites are transfinite" requires AC? (I'm intrigued by your statement about ordinals and AC -- this is something I need to investigate much more -- but I'm just asking about cardinals at the moment.) --Michael C. Price talk 07:36, 26 July 2006 (UTC)
Look, there is simply no way that Cantor was intending to make the infinite/Dedekind-infinite distinction when he coined the word "transfinite". He would, I think, have found completely convincing the informal proof that all infinite sets are Dedekind infinite, and would not have understood you if you pointed out that it "uses AC"; his approach to proof was informal rather than axiomatic.
As for "infinite" being "more inclusive" than "transfinite"—what I'm saying is that Cantor intended "transfinite" to be a weaker notion than (absolute) infinite. A transfinite aleph is pretty big, but it's not as big as an absolutely infinite collection. Dedekind infinite, on the other hand, is (in the relevant context) stronger than infinite. --Trovatore 15:54, 26 July 2006 (UTC)
Sorry, I don't see that what Cantor intended is relevant; I'm sure he would have thought that the (in)finite/Dedekind-(in)finite distinction was bogus, but we know better and that is relevant. Please don't start responses with "Look", OK? --Michael C. Price talk 17:52, 26 July 2006 (UTC)
Cantor invented the term, and it was adopted by mathematicians generally in a way that more-or-less reflected Cantor's understanding. Of course linguistic usage is subject to change; if it were the case that mathematicians today followed Suppes' usage more than Cantor's, then we would be writing primarily about Suppes' usage. But that change has not happened—I do not believe most mathematicians using the word "transfinite" use it to mean specifically "Dedekind infinite". Rather, they use it to mean "infinite", when talking about the sorts of objects Cantor considered (rather than other sorts of infinite quantity).
Suppes is an important enough source that this article should mention the usage, but that's about all, I think. The unambiguous and universally-understood term is "Dedekind infinite", and we have an article for that, which is where the substantive discussions of that concept should go. --Trovatore 18:02, 26 July 2006 (UTC)
Let me try to summarise what we can, perhaps, both agree on, and you can tell me if I've "got it" yet. Cantor introduced the term "transfinite" as a weaker concept than "infinite", to express the notion that some numbers existed beyond the finites but lacking some of the pathological qualities associated (at the time) with full-blown infinity. Now, however, whenever we distinguish between "infinite" and "transfinite" -- which isn't often -- we identify "transfinite" with "Dedekind infinite". Consequently transfinite becomes a stronger concept than infinite, because "Dedekind infinite" is a stronger concept than "infinite". With the widespread adoption of AC these distinctions have become irrelevant for most mathematicans. --Michael C. Price talk 13:16, 27 July 2006 (UTC)
No, I don't agree with the second part, that starts "now, however". If I were to use the terms distinctively, the most likely reason would be that I was distinguishing the transfinite objects Cantor considered (ordinals and cardinals) from other sorts of infinite quantity, such as the +∞ of the extended real number line, or infinite surreal numbers or hyperreals. I think the distinctive usage you propose is not very standard and risks misunderstanding. --Trovatore 15:42, 27 July 2006 (UTC)

I've read the preceding conversation. What I have taken from it is this: Suppes defines "transfinite" to mean "Dedekind infinite" and "infinite" to mean "infinite" (of cardinals, bigger than any finite). I agree with Trovatore that this is nonstandard (though I'm sure Trovatore can tell you that I'm not a set-theorist). Nevertheless, we can add a 5-word footnote along the lines of

Some authors (e.g. Suppes) use the word "transfinite" to mean Dedekind infinite.

Any questions of AC belong in that article (Dedekind infinite), but I believe that issue is already treated there. For Suppes, AC makes transfinite and infinite mean the same thing. For the rest of us, AC means that infinite and Dedekind infinite are the same thing. For the rest of us, it has nothing to do with the word "transfinite", so does not belong in this article. The word "transfinite" was invented by Cantor to distinguish a more precise notion of infinite. It may have been a useful neologism in Cantor's day, but it is mostly redundant today, Suppes' nonstandard definition notwithstanding. It would also probably be good if some others of us had a look at the definitions in Suppes. -lethe talk + 15:47, 27 July 2006 (UTC)

Let's not forget that Arthur said
In set theory, transfinite is often used as synonymous to "Dedekind infinite"
so that Suppes' definitions are obviously not that non-standard. --Michael C. Price talk 01:27, 28 July 2006 (UTC)
With all respect for Arthur, I don't agree with his characterization there. My PhD is in set theory (as is Arthur's) and I have not come across this usage, at least on any regular basis. In fact, it would strike me as very odd to call any non-wellorderable set "transfinite", whether Dedekind infinite or not, so strongly is the term connected with the Cantorian ordinals and alephs, and with transfinite recursion and transfinite induction (both of which are wellordered processes). --Trovatore 06:15, 28 July 2006 (UTC)

How about this -- we define "transfinite ordinal" as "infinite ordinal"; and define "transfinite cardinal" as "infinite well-orderable cardinal"? Then transfinite cardinals would all be Dedekind-infinite; but in some models there could be Dedekind-infinite cardinals which were not transfinite. JRSpriggs 06:52, 28 July 2006 (UTC)

Are they sourced definitions? --Michael C. Price talk 07:06, 28 July 2006 (UTC)
No. They just represent my sense of that at which we are getting in this discussion. Cantor assumed the axiom of choice. So for him all cardinals were well-orderable. Now we recognize the possibility of models in which there are cardinals which are not well-orderable. But he did not mean to apply "transfinite" to them, so why should we? The other obvious generalization would be "infinite cardinal", but that might include Dedekind-finite infinite cardinals which Suppes and Arthur Rubin do not wish to include. JRSpriggs 07:28, 28 July 2006 (UTC)
That might possibly count as original research. Whatever definition(s) we decide on I think there must be sufficient context so that anyone coming to the page can relate to whatever they've heard or read elsewhere. From what I can gather "transfinite" means different things to different people so no single, OR definition is the solution. I think we have to give more of the historical background, as we've discussed here, in the article for context. --Michael C. Price talk 07:39, 28 July 2006 (UTC)

### Sources to definitions

• Jean E. Rubin, Set Theory for the Mathematician, Holden-Day (San Francisico, 1967), Library of Congress # 67-13848, Definition 12.2.1 (p. 296)
m is transfinite =Df ${\displaystyle \mathbf {m} \geq \aleph _{0}.}$
• H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice, II (oh, look it up somewhere else), Definition 0.13(d), p. xxvi
A cardinal κ is transfinite if ${\displaystyle \aleph _{0}\leq \kappa .}$

(I suspect my parents' other books on the axiom of choice use the same definition; this is just the only one I have here at the moment.)

• Herbert B. Enderton, Elements of Set Theory, Academic Press (1977), ISBN 01223844407, p. 14
(Refers to Cantor's work, in an historical context, without a specific definition of "transfinite numbers".)

This concludes the list of all the set theory books that I have here which have "transfinite" (not in the context of "transfinite induction" or "transfinite recursion") in the index. Anyone else? — Arthur Rubin | (talk) 14:59, 28 July 2006 (UTC)

For completeness:

• Patrick Suppes Axiomatic Set Theory Dover, 1972, ISBN 0486616304, page 155, definition 27:
${\displaystyle \mathbf {m} }$ is a transfinite cardinal if and only if there is a Dedekind infinite set A such that ${\displaystyle \kappa (A)=\mathbf {m} }$

and offers the following equivalences:

Theorem 58, page 155

${\displaystyle \mathbf {m} +1=\mathbf {m} }$

Theorem 64(i), page 157

${\displaystyle \aleph _{0}\leq \mathbf {m} }$

Theorem 64(ii), page 157

there is a cardinal ${\displaystyle \mathbf {n} }$ such that ${\displaystyle \aleph _{0}+\mathbf {n} =\mathbf {m} }$

--Michael C. Price talk 19:15, 28 July 2006 (UTC)

So in case anyone wasn't paying close enough attention, let me point out that Arthur is not citing Enderton in support of the "Dedekind infinite" usage. So far the sources we have for that are Suppes, and Arthur's mom and dad.
I guess my biggest objection to this usage is that it doesn't give the listener fair warning that you're stepping out of the default context, in which full AC holds, into a context where you're not sure even of countable AC and where you might have radically pathological objects like infinite but Dedekind-finite sets. As the word "transfinite" was historically used, it carried no such implication. As I say, language could certainly change on this point, but I don't believe that it has done so, except among a minority of workers. When you say "Dedekind infinite", OTOH, you are giving fair warning (otherwise, why wouldn't you simply say "infinite"?). --Trovatore 15:21, 28 July 2006 (UTC)
That this usage doesn't depend on AC is to its advantage -- which is presumably why it is the preferred precise, documented definition. Only documented, verifiable definitions should appear in Wikipedia. --Michael C. Price talk 08:15, 30 July 2006 (UTC)
(Aside to Arthur: Is "Dedekind infinite" really equivalent to embedding ${\displaystyle \aleph _{0}}$, over ZF alone? I see the right-to-left implication, but not the left-to-right.) --Trovatore 15:21, 28 July 2006 (UTC)
(edit confict)
Answer to aside; yes; Dedekind infinite#Dedekind-infinite sets in ZF is correct.
Specifics; if f is a surbijection from A onto a proper subset B of A, select an element a in A but not in B. Then the map g from ω to A defined by g(n)= fn(a) (the nth iterate of f applied to a) can be shown to be an injection.
On the other hand if g from &omega to A is an injection, then we can define: f(g(n)) = g(n+1), and f(a)=a if a is in A but not in the range of g. Then f maps A onto A \ {g(0)}.
Also, you're correct that I'm not using Enderton to support my assertion; but it seems to read more along the line that "Georg Cantor used the term "transfinite numbers" to mean....", not really implying that it's used that way today. — Arthur Rubin | (talk) 19:20, 28 July 2006 (UTC)

To Arthur: I think you mean "injection" rather than "surjection". Otherwise, a counter-example would be A={a,b}; B={b};f={<a,b>,<b,b>}. Then f is surjective from A onto B with B a subset of A, but g(0)=a; g(n+1)=b for any n. Not what I would call Dedekind-infinite or any kind of infinite. Changing gears, what about a set which is the union of ω with a Dedekind-finite infinite set? It is Dedekind-infinite, but quite pathological. Do you want its cardinality to be "transfinite"? JRSpriggs 03:38, 29 July 2006 (UTC)

(Fixed "thinko" above.) I'm afraid so. There are models of set theory in which ${\displaystyle \aleph _{1}}$ is not less than or equal to ${\displaystyle \beth _{1}}$; we still want the cardinality of the reals to be transfinite. — Arthur Rubin | (talk) 18:37, 29 July 2006 (UTC)
Well, that's kind of a non sequitur; you haven't argued there are models where the reals are the union of a countable set with a Dedekind-finite set. (An interesting question on its own; not sure of the answer offhand.)
But more on topic, I simply don't buy that the usage you're promoting has any real traction among the bulk of set theorists. It doesn't, in my experience. If you ask most set theorists if the reals are transfinite even if they can't be wellordered, they'd probably scratch their heads, unsure what's really being asked, and say "yeah, I guess". But they'd probably do the same with regard to infinite Dedekind-finite sets.
Bottom line, if you want to be understood, you say "Dedekind infinite", which is a perfectly good word; why do we want to force "transfinite", counter-historically, to serve this function? Not, of course, that it's our choice; if this usage had become the standard one, we'd use it. --Trovatore 19:01, 29 July 2006 (UTC)
No, the bottom line is that the Dedekind infinite definition is documented and verifiable. Unverifiable claims are irrelevant to Wikipedia and should not appear. --Michael C. Price talk 07:57, 30 July 2006 (UTC)
It's a minority usage. Yes, we should mention it here. But the substantive discussion belongs at Dedekind-infinite set, which is the standard name. --Trovatore 16:32, 30 July 2006 (UTC)
Logically there's nothing to choose between here or at Dedekind infinite since the terms are equivalent. With regards to your claim to it being a minority usage, where is that documented? --Michael C. Price talk 19:31, 30 July 2006 (UTC)
We don't need to state in the article that it's a minority usage, so we don't need to document it. I think Lethe's solution is fine, or I'd go as far as saying we could have a brief paragraph, something like
Some authors, for example Suppes, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold.
The reason that any further content relating to this usage belongs at Dedekind-infinite set is twofold: It's the standard term, and it's unambiguous. Note that the second reason sometimes controls even when opposed to the first; take a look at indicator function and characteristic function. Really almost everyone uses the term "characteristic function" for what probabilists call an "indicator function", but we have our article at indicator function because it's just too awkward to do it any other way. I'm not entirely happy with the solution there, but in the case currently under discussion we don't have to choose; the two considerations both point in the same direction. --Trovatore 19:45, 30 July 2006 (UTC)
OK, proposed paragraph looks uncontentious. Let's go with it. Possibly we could add Suppes' and the Rubins' equivalent definitions as well, but let's go with what we can all agree with for now. --Michael C. Price talk 07:08, 31 July 2006 (UTC)

## Encyclopaedic?

Mathematical concepts are abstract in the extreme and therefore inherently hard to explain, but is there not perhaps a more "hierarchical", way to define them so a lay reader can come to some simple understanding? At present the article in almost entirely incomprehensible to a non-mathematics student. Is not the aim of wiki to make information accessible? Could anyone insert a clearer, plain English definition, to explain the term to non-mathematicians? If something like this cannot be accomplished these maths entries, erudite though they may or may not be, become "bookish" and unhelpful to an Encyclopaedia's aim. LookingGlass (talk) 06:01, 30 April 2008 (UTC)

Seconded. I don't expect anyone to be able to make this page instantly understandable to people who haven't studied advanced math (e.g. me), but I think a better attempt could be made. Right now, I've got that a transfinite number is somewhere between finite and infinite numbers. Whatever that means. -person, Dec. 9 —Preceding unsigned comment added by 169.229.109.115 (talk) 06:24, 10 December 2008 (UTC)
Actually, "transfinite" means the same as "infinite". As the lead says, Cantor renamed it because he wanted to avoid some of the prejudices which were attached to the word "infinite". For example, the presumption that all infinities are the same. JRSpriggs (talk) 19:57, 11 December 2008 (UTC)

## Disturbing formulation

In the article it is said:

"If the axiom of choice holds, the next higher cardinal number is aleph-one, {\aleph_1}. If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-zero."

I would assume: If not, there is no well defined aleph-one as there may be many larger cardinal numbers which are mutually incomparable and it makes no sense to speak of aleph-one at all. So, I think the second sentence should rather read something like "If not, there may be no unique cardinal larger than aleph-zero." (Which is still not good, since it could be read to imply there could be no higher cardinal at all, which is definitely not the case, but I have no better idea currently...) 134.169.77.186 (talk) 15:51, 21 January 2009 (UTC) (ezander)

I'm afraid not. Aleph One can be defined without the axiom of choice in ZF set theory (or even ZFU). In fact, the entire Aleph hierarchyα for ordinal numbers α can be defined in ZF or ZFU. — Arthur Rubin (talk) 16:09, 21 January 2009 (UTC)
Elaborating a little — you don't need AC to prove that there's a set of all countable ordinals. ${\displaystyle \aleph _{1}}$ is the cardinality of that set. If AC fails, then there could be cardinalities incomparable with ${\displaystyle \aleph _{1}}$, but there's still an ${\displaystyle \aleph _{1}}$, and while it wouldn't in that case be the unique smallest cardinal greater than ${\displaystyle \aleph _{0}}$, it would still be a minimal cardinal greater than ${\displaystyle \aleph _{0}}$. That is, there would be no cardinal strictly in between. --Trovatore (talk) 20:30, 21 January 2009 (UTC)

## Surreals

Over in the surreal number article, it is claimed that surreals are a superset of the transfinite ordinals. I note that we don't link to that article from this one. Is there some debate about the validity of the surreals, or what is going on? Ethan Mitchell (talk) 19:59, 14 August 2011 (UTC)

It's accurate, but the relevance of the surreal numbers to conventional ordinal numbers and set theory is in question. — Arthur Rubin (talk) 21:00, 14 August 2011 (UTC)

## Restricted to ordinals and cardinals

The terms “transfinite number” and “infinite number” are much more generally used, for example for additional numbers in non-standard models of arithmetic, hyperreal numbers or infinite-values in measure theory etc. --Chricho ∀ (talk) 16:05, 25 December 2011 (UTC)

It's true that infinite number may refer to a nonstandard element of such a model. But transfinite number — no, I don't think so. I am unaware of any such usage. --Trovatore (talk) 16:08, 25 December 2011 (UTC)
However, “infinite number” redirects here. --Chricho ∀ (talk) 23:16, 23 January 2012 (UTC)
Ah. That's the sort of thing that's dealt with via hatnotes. Or maybe retargeting the redirect — I think infinity might be a better target. --Trovatore (talk) 23:42, 23 January 2012 (UTC)

## "Absolute infinity"

There is a tendency to think of 'infinity' as having, however unapproachable and intangible, some fixed numerical value. Surely this is exactly what Cantor sought to dispel in inventing the important and meaningful term 'transfinite'. To imply that an infinite series of infinities must culminate in a so-called "absolute infinity" is just what he avoided. Having established mathematically that infinity (aleph null) when multiplied by itself is unchanged numerically (etc), and proceeding to establish his infinite series of such infinities, he made it clear that by definition nothing exceeds ANY infinity. And hence, although there are an infinity of DIFFERENT infinities, they cannot be arranged in an order of size. Thus showing the transfinite numbers to be quite unique in behaviour as compared with cis-finite numbers. It has also been established that there are different kinds of zero, but no-one would attempt to arrange them in order of nullity.125.239.240.174 (talk) 20:02, 23 January 2012 (UTC)

You have been able to find a mathematician or philosopher who would agree with you in the 19th century, but not now. There is no sense in which (aleph 0), the cardinality of the natural numbers (or integer, rational, algebraic, constructable) numbers is not less than (beth 1), the cardinality of the reals. — Arthur Rubin (talk) 21:55, 23 January 2012 (UTC)

If 'infinity' is defined as "That number other than which no number is greater", the notion that the series of transfinite numbers can be graded in size order fails. Being outside the mathematical rules such as n+x or n.x are >n, transfinite numbers are not governed by denumerable/quantifiable bounds.They HAVE no size. The notion that the transfinite/infinite must be 'huge' is fallacious; they are of different types, but might just as reasonably thought of as quite small. This was postulated millennia ago by Plato in "Timaeus", where the example of eternity 'small enough to fit in the palm of the hand' is represented by eternity as a small gold coin, for which change is being paid out in such tiny base-metal coin that the process takes forever. Just as time is not a linear series of moments, nor a line one of positions, so infinity is not the culmination of a series of units.125.239.241.123 (talk) 01:13, 15 February 2012 (UTC)

My understanding is that Absolute Infinity was a somewhat theological notion to Cantor — not theological in the somewhat pejorative sense applied by narrowminded mathematicians to mathematics they consider insufficiently concrete, but literally theological, as in having to do with God. Cantor (again this is my interpretation) held that transfinita ordinata could be conceived of by God, and therefore must exist in the mind of God, but that the collection of things conceivable by God (an "inconsistent multiplicity") was absolutely infinite, not merely transfinite. --Trovatore (talk) 22:10, 23 January 2012 (UTC)
I don't know whether that's accurate; regardless, it has no relevance to this article. — Arthur Rubin (talk) 22:33, 23 January 2012 (UTC)
I don't entirely agree. Its relevance is somewhat tangential, and mainly regards the origin of the word rather than the mathematical content of the article, but it is not quite irrelevant. --Trovatore (talk) 22:35, 23 January 2012 (UTC)

@Trovatore Any sources? @IP “And hence, although there are an infinity of DIFFERENT infinities, they cannot be arranged in an order of size.” No, thanks to the well-orderning theorem (equivalent to the axiom of choice) we can arrange all finite and infinite cardinalities of sets. --Chricho ∀ (talk) 23:20, 23 January 2012 (UTC)

A quick Google Scholar search picks up this article, which I haven't read (costs \$36; I'll consider buying it as I think I'd like to read it). Probably working a little harder on the search it would be possible to find something gratis. --Trovatore (talk) 23:34, 23 January 2012 (UTC)

## Set Theory for the Mathematician

My mother's book was based on von Neumann–Bernays–Gödel set theory, not Morse–Kelley set theory. I don't want to make the change, myself, as I may eventually get royalties from the book, making this a potential WP:COI violation. — Arthur Rubin (talk) 16:18, 25 October 2014 (UTC)

According to the Zentralblatt Math review, the author uses "Morse's modification of the von Neumann–Bernays–Gödel axiomatic system (modified further such that atoms are allowed)".[1] Is that inaccurate? The easiest will be to replace the sentence by "Uses an axiomatic approach", without identifying the specific axiomatic system. Would that do? (Aside: the review by Shepherdson, J. C. (November 1969), Bulletin of the London Mathematical Society, 1 (3): 437–438, doi:10.1112/blms/1.3.437CS1 maint: untitled periodical (link), writes "his proof", apparently meaning "the author's proof".)  --Lambiam 10:25, 24 May 2021 (UTC)

## Quote

The assumption that apart from the absolute - unreachable by any determination - and the finite, no modifications can exist which, though not finite, are nonetheless determinable by numbers and consequently are what I call the proper infinite; this assumption I do not find justified by anything [...]. What I maintain and believe I have proved in this paper as well as in earlier attempts is that after the finite there is a transfinitum (which could also be called suprafinitum), i.e., an unlimited gradation of definite modes which in their nature are not finite but infinite, yet which, just as the finite, can be determined by definite, well-defined and mutually distinguishable numbers. (Cantor 1883b, p. 176, Grundlagen)

To me this seems kind of at odds with the description in the lead?:

"Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite."

--Fixuture (talk) 22:35, 4 July 2015 (UTC)

## New examples section

The new section is invalid because the Wolfram article (without necessary references) it is copied from is invalid, per all ordinal infinities fitting in aleph0. The reference to Cantor does not link to Cantor. The editor thinks it follows Wikipedia policy since it is from Wolfram, however the author of the piece is not known and even it was Wolfram himself would still not follow Wikipedia policy. The Wikipedia section also leaves out critical stuff. Victor Kosko (talk) 08:20, 4 June 2020 (UTC)

This section was removed by and is visible in this version It's not at all due to Wolfram, I think it's due to Cantor. (I'm not sure and don't know how to check). Its certainly a one paragraph summary of about 3 or 4 chapters from John Horton Conway's book On Numbers and Games, from 1976. Conway did not invent this stuff either; he's just recapitulating older results and placing them in a new, much improved framework. I'm going to restore that content, and replace the reference with one to Conway's book. If you don't like it, well, I'm not married to this stuff. I just thought that Conway's book was rather entertaining. 67.198.37.16 (talk) 23:32, 26 May 2021 (UTC)
BTW (Part of) what Conway does is takes results from Berlekamp(?), and shows how to naturally define ${\displaystyle \omega -1}$ and proves that this is (a) a valid number, and (b) is strictly less than ${\displaystyle \omega }$. He is then able to recursively reach ${\displaystyle \omega /2}$, ${\displaystyle \omega /4}$ and so on, and eventually ${\displaystyle \omega /\omega }$, which is still a number, which is strictly greater than zero, strictly less than ${\displaystyle \omega /n}$ and incomparable to any natural number. (It is impossible to determine the truth or falsity of any inequality involving ${\displaystyle \omega /\omega }$ and a natural number or a rational number. I think this holds in the sense of "decidability" as in Turing computatbility, but I may be wrong about that. Basically, the determination of great-than/less-than requires infinite recursion -- an infinite loop. Whoops.) This is just a tiny taste of all the fun things he's able to do. all this is possible because he has axiomatic definitions for subtraction and division and inequality, and not just addition and multiplication. These axiomatic definitions can also be applied to two-player games, both finite and infinite. That's where the fun reaches a whole new level. 67.198.37.16 (talk) 00:03, 27 May 2021 (UTC)
I totally agree with you that this stuff is cool. What I don't see is that it has any place in this article. Surreal numbers are great, but they aren't "transfinite numbers" in the usual sense.
I'm going to indulge myself to correct a couple points of fact in the above; it's not strictly according to the rules but given that it's there I don't like to leave it unaddressed. The attribution to Berlekamp is interesting; I've never heard of that but can't comment either way on accuracy. In the surreals, yes, ${\displaystyle \omega -1}$ is a "number" (that is, surreal number) less than ${\displaystyle \omega }$. However, in the more usual sense, ${\displaystyle \omega }$ is an ordinal, not a surreal, and ${\displaystyle \omega -1=\omega }$.
In the surreals, ${\displaystyle {\frac {\omega }{\omega }}=1}$. This follows from the fact that the surreals form a field.
I'm pretty sure the surreals are totally ordered; they have no incomparable elements.
Finally, it is not possible for a single statement to be "undecidable" in the sense of Turing computability. That sort of decidability applies only to infinitely many statements (e.g. there is no fixed program that can decide whether an arbitrary Turing machine halts, but for a fixed Turing machine, it either halts or it doesn't; it can't be "undecidable" in that sense, though it might be independent of some formal theory you're relying on). --Trovatore (talk) 18:57, 28 May 2021 (UTC)

## Surreals and hyperreals as "analogous transfinite" extensions of the reals

So the reference David Eppstein added does indeed use the word "transfinite" in the title, but as part of the phrase "transfinite function iteration". I do not have the reference to look at, but just from the title, this sounds to me as though a function is being iterated transfinitely; that is, up to a transfinite ordinal. That does not make the surreals themselves "transfinite", and it doesn't establish an analogy between the surreals' extension of the reals to the transfinite numbers' extension of the natural numbers.

The other consideration here is that we need to distinguish carefully among the various meanings of "transfinite". You can see from discussions above that the (in my opinion ahistorical) extension to mean "Dedekind infinite" has some currency, but this must not be conflated with the original meaning. --Trovatore (talk) 20:51, 27 May 2021 (UTC)

Update: I found the ref. It does mention (once) "transfinite surreals", but it looks like a nonce term. It never explicitly defines a "transfinite surreal" as a distinct concept. In context it seems to mean surreals that have length greater than ω in a particular representation. Absent evidence that this phrase has broader currency with this specific meaning, I don't think that's enough cause to claim to talk about surreals as "transfinite" in this article. --Trovatore (talk) 02:20, 28 May 2021 (UTC)