Talk:Transfinite number

message: 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)
thread_title: presentation method

message: Shouldn't this be merged with infinite 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)
thread_title:  merge? 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Encyclopaedic?

message: 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)
replies: text: 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 hierarchy ℵα for ordinal numbers α can be defined in ZF or ZFU.  — Arthur Rubin  (talk) 16:09, 21 January 2009 (UTC)
replies: text: Elaborating a little — you don't need AC to prove that there's a set of all countable ordinals.  \aleph_1 is the cardinality of that set.  If AC fails, then there could be cardinalities incomparable with \aleph_1, but there's still an \aleph_1, and while it wouldn't in that case be the unique smallest cardinal greater than \aleph_0, it would still be a minimal cardinal greater than \aleph_0.  That is, there would be no cardinal strictly in between. --Trovatore (talk) 20:30, 21 January 2009 (UTC)
thread_title:  Disturbing formulation 

message: 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)
replies: text: 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)
thread_title:  Surreals 

message: 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)
replies: text: 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)
replies: text: However, “infinite number” redirects here. --Chricho ∀ (talk) 23:16, 23 January 2012 (UTC)
replies: text: 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)
thread_title:  Restricted to ordinals and cardinals 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text:  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)

text: 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)
thread_title:  "Absolute infinity" 

message: 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)
replies: text: 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)". 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 , writes "his proof", apparently meaning "the author's proof".)  --Lambiam 10:25, 24 May 2021 (UTC)
thread_title:  ''Set Theory for the Mathematician'' 

message: 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)
thread_title:  Quote 

message: 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)
thread_title:  Surreals and hyperreals as "analogous transfinite" extensions of the reals 

message: The interesting thing is that a countably infinite number of elements can be arranged in such a way as to create an uncountably infinite number of sets.
For example, the real numbers between 0 and 1 are uncountably infinite, yet the number of individual digits needed to express them is countably infinite. 2600:4040:5D30:4800:1D47:CB3:1139:3583 (talk) 14:48, 28 November 2022 (UTC)
replies: text: If all real numbers betweem 0 and 1 were written down in decimal notation, each digit would occur uncountably often. - Jochen Burghardt (talk) 15:40, 28 November 2022 (UTC)
replies: text: No. You can construct them in such a way as to be able to count the digits.  2600:4040:5D30:4800:1D47:CB3:1139:3583 (talk) 16:43, 28 November 2022 (UTC)
replies: text: I don't think anyone here has any idea what you're talking about it seems like you need only ten "individual digits" to express them all in base ten, and you could reduce that number to two by an obvious expedient but in any case it doesn't seem on-topic for this talk page, unless you're proposing a change to the article. --Trovatore (talk) 18:12, 28 November 2022 (UTC)
thread_title:  paradox? 

message: I realize this is hardly the responsibility of wikipedia, but it may be prudent to note that epsilon is used in two different respects in related articles, namely in the infinitesimals https://en.wikipedia.org/wiki/Infinitesimal and the transfinites https://en.wikipedia.org/wiki/Epsilon_number 137.118.200.159 (talk) 01:00, 19 January 2023 (UTC)
replies: text: All through mathematics you have the issue of the same symbol being used in different ways.  I don't see any need to call out this particular collision in the article. --Trovatore (talk) 02:08, 19 January 2023 (UTC)
thread_title:  Unclear notation