Talk:Absolute infinite

I'm not at all happy with this page. I don't know a lot about transfinites but there are things here that worry me. Limit? Not in the normal sense. This needs expansion at least. You can't just give something a name and hope that it won't introduce inconsistency to do so. Often it does. Going to have a think about this. Andrewa 10:50 16 Jul 2003 (UTC) This is a proper noun. I have moved the page back to "Absolute Infinite" (with "absolute infinite" redirecting here). -- The Anome 08:55, 18 Sep 2003 (UTC) In the see also section, what does "The Absolute" refer to?? Jaberwocky6669 02:28, Mar 30, 2005 (UTC)

Allow me to redo my question, what is the "absolute" at the top of the article? Jaberwocky6669 02:30, Mar 30, 2005 (UTC)

The sentence " Indeed, naive set theory might be said to be based on this notion. " in Burali-Forti is vague and seems incorrect to me. For example, in ^[1] the treatment is fully consistent with ZFC. I suggest removing this sentence altogether as it does not add anything useful and confuses the inexpert reader. cerniagigante (talk) 16:34, 24 January 2017 (UTC)

Is this serious? Did Cantor actually believe that "that every property of the Absolute Infinite is also held by some smaller object"? I mean, he was clearly an incredibly smart guy, but on the face of it, that's an idiotic opinion.

For example, it implies that there is some smaller object that is also larger than all objects besides itself. -Rwv37 04:25, Jun 27, 2005 (UTC)

Isn't the universe such an object? 24.174.45.155 21:33, 16 July 2006 (UTC)

No. The universe is an object that is larger than all objects besides itself. But that property is *not* shared with some smaller object. -Rwv37 00:41, 22 July 2006 (UTC)

I am using this exchange as justifying this article as "unclear." Cantor's quote "that every property..." is almost certainly a reference to the Reflection Theorem of set theory--but I can't be sure without a citation. See Reflection principle for more information. Cobaltnova 22:14, 10 November 2007 (UTC)

The notion of multiplicity Cantor describes here can't possibly be the same as on multiplicity A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a sequence. 24.174.45.155 21:33, 16 July 2006 (UTC)

I think he means what we now call a cardinal number. I just piped the wikilink which I hope is ok. 75.62.4.229 (talk) 04:37, 24 November 2007 (UTC)

Close. I am virtually certain he just means a set. Because the definition of a well-ordered set is one in which each subset has a least member. Hccrle (talk) 21:36, 29 August 2009 (UTC)

I followed the link and was surprised to arrive at Cardinal number which seemed unhelpful. I therefore added an Other senses section to multiplicity (mathematics) and linked to it. There I said he seemed to mean an ordered set, but it would be helpful if someone with access to the original and translation could check what he meant and provide an exact definition and explanation, or a link to (part of) a different article. PJTraill (talk) 11:31, 6 November 2022 (UTC)

My edit there was reverted on the grounds that Wikipedia is not a dictionary and that I was just providing the common-language meaning, but that seems inaccurate. PJTraill (talk) 11:59, 6 November 2022 (UTC)

I started a topic at Talk: Multiplicity (mathematics) to discuss this further —— so doing so here seems redundant! PJTraill (talk) 12:17, 6 November 2022 (UTC)

I have removed the link and added a note linking to set (mathematics). PJTraill (talk) 14:46, 6 November 2022 (UTC)

The way that leads from Infinite to Absolute Infinite could lead also from Absolute Infinite to Absolute Absolute Infinite, from Absolute Absolute Infinite to Absolute Absolute Absolute Infinite and beyond. So, where is the limit? 89.1.112.168 07:25, 13 September 2007 (UTC)

The limit is at the beginning of your diatribe: the Absolute Infinite is the order of proper classes, which are not sets. Therefore we are not allowed to form bigger sets. So there is nothing larger than the Absolute Infinite. That's what is absolute about it. There is no Absolute Absolute Infinite. Hccrle (talk) 22:00, 29 August 2009 (UTC)

Absolute Infinity is a big Number but not the biggest you can just add 1 to make it Absolute Infinite +1 2601:644:4181:B550:2CAA:57E9:6851:F4F6 (talk) 20:37, 4 December 2024 (UTC)

I am removing "mysterious" from the description of proper classes. These ideas are mathematically well-defined (see Kunen, Kenneth "Set Theory: An Introduction to Independence Proofs").

I think the corresponding idea in axiomatic set theory is the von Neumann universe also known as the cumulative hierarchy. 75.62.4.229 (talk) 04:39, 24 November 2007 (UTC)

You're on the right track. Assuming the Axiom of Choice, the transfinite cardinal numbers are ordinal numbers that form the backbone, so to speak, of the von Neumann universe. Every stage of construction of that universe has a rank, which is an ordinal number and is a member of that stage. The class of all the ordinal numbers is a proper class, and is the Absolute Infinite. Hccrle (talk) 08:19, 30 August 2009 (UTC)

I shall call him XIBTAI - acronym of Xibtai Is Bigger Than Absolute Infinite! —Preceding unsigned comment added by 89.0.54.122 (talk) 08:27, 11 September 2008 (UTC)

See the above section Absolute Absolute Infinite ... Hccrle (talk) 22:05, 29 August 2009 (UTC)

So, what is larger than Nothing? 217.132.68.201 (talk) 11:18, 16 February 2010 (UTC)

Currently the article uses this translation:

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

Jané however has this translation in "The role of the absolute infinite in Cantor's conception of set":

The actual infinite can be divided according to three aspects: first, as it is realized in the supreme perfection, in the completely independent, extrawordly being, in God, where I call it absolute infinite or simply absolute; second as it is represented in the dependent world of things created; third as conceived in abstracto as a mathematical quantity, number or ordertype. (Cantor 1887-88, p. 378)

And from the original quote (https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf) it seems to be:

The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extrawordly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or ordertype. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.

--Fixuture (talk) 23:45, 4 July 2015 (UTC)

That is a major difference. The first two translations suggest that Cantor was naive; the last seems suggests that Cantor's thinking was quite modern and up-to-date (viz a premonition of 20th century transfinite work). 67.198.37.16 (talk) 04:27, 8 July 2016 (UTC)

So I took your word for it and replaced that translation with the 3rd one. Thanks for your review. I included the original quote in German in the reference. --Fixuture (talk) 21:44, 18 July 2016 (UTC)

it's called Absolutely Absolute Infinite.89.0.54.122 (talk) — Preceding unsigned comment added by 178.90.121.55 (talk) 06:16, 21 August 2023 (UTC)

@178.90.121.55 I don't know what you're talking about. do you have evidence? 2601:83:4280:A9E0:282E:701:B11:EFF4 (talk) 19:15, 21 August 2023 (UTC)

10^{absolute infinite} — Preceding unsigned comment added by 178.89.254.140 (talk) 12:12, 8 September 2023 (UTC)

How 'bout Ω^Ω? Guess like zero it's "undefined". P.I. Ellsworth , ed. ^{put'er there} 12:27, 13 May 2024 (UTC)

File:Australia in its region (Ashmore and Cartier Islands special).svg — Preceding unsigned comment added by 95.56.95.231 (talk) 18:57, 10 September 2023 (UTC)

What is the source supporting the information added (with no supporting source) in August 2025 by @Farkle Griffen regarding the Hebrew letter serving as the symbol for absolute infinity and removing information about the actual symbol Ω? -- Prokurator11 (talk) 03:24, 7 December 2025 (UTC)

I've removed the tav stuff. If re-added we should get a source and we should explain who uses it that way. I don't think it's common.

As for Ω, that does seem to have been used in Cantor's original writings, but I don't think it's much used these days. I wouldn't put it in the lead sentence. In my experience, more common names would be ORD or ON (the latter for "ordinal numbers").

I see that a discussion of proper classes was removed a while back. That may have been the correct thing to do given the text that was there, but I imagine we can find sources that discuss the connection. --Trovatore (talk) 22:35, 7 December 2025 (UTC)

The corresponding link on the page about the letter itself still remains. ~2025-39232-91 (talk) 07:00, 8 December 2025 (UTC)

@Prokurator11, @Trovatore

Cantor defined ${\displaystyle \Omega }$ as the "set" of all ordinal numbers, and ת‎ as the "set" of all cardinals. It is this ת‎ that Cantor identified as "Absolute infinite", though later in his writing he would often use "Absolute infinite" the specific class of cardinals, and "Absolute infinite", a more abstracted sense of size, interchangeably. It's hard to find good, free English sources on this since Cantor never really talked about the Absolute infinite in any formal, non-theological context, and most of the primary sources on it are private letters between Cantor and Dedekind. These two sources are pretty good though:

One of them was related to the idea of the totality of all cardinal numbers. [...] He called this set ת‎, or absolute infinity, which means ת‎ = ${\displaystyle \aleph _{0},\aleph _{1},\aleph _{2}...}$

- Infinity: New Research Frontiers, pp. 40-41

And also this book, p 243-246

The symbol ${\displaystyle \Omega }$ as "the symbol" for absolute infinite was only introduced in 2017 by an IP editor, and was not removed. This seems to have done real damage, since, a series of youtube videos gaining hundreds of thousands of views each, and which apparently became part of a googology fandom, has been spreading the incorrect symbol. – Farkle Griffen (talk) 19:48, 8 December 2025 (UTC)

@Farkle Griffen, it would be helpful if, rather than adding unsourced information to the preamble that already lists other sources that don't actually support it (which can give the impression of misrepresenting sources), you provided the supporting sources upfront. I don't know this topic well enough to judge, so if other editors familiar with it consider your source authoritative, I'm happy to see the information restored with proper citation. -- Prokurator11 (talk) 02:54, 9 December 2025 (UTC)

Honestly it's very strange that he would prefer the class of cardinals over the class of ordinals as a way of representing absolute infinity, but that may be with hindsight. If there were a cardinality representing absolute infinity it would be the ordinals, not the cardinals.

In any case I think the article should not emphasize equating absolute infinity with a particular mathematical object. I think it's more productive to discuss the philosophical and theological connections. --Trovatore (talk) 05:30, 9 December 2025 (UTC)

I looked up Farkle's first link. It's in a collection edited by W. Hugh Woodin, which is pretty impressive, but it's in an article written by Wolfgang Achtner, who was not a mathematician,UPDATE: His bio does say he earned an additional degree in mathematics; it doesn't give specifics which may partially explain why the section in question is a little hard to read. Achtner says that Cantor used ת to represent the "set of transfinite sets", says that Cantor called this absolute infinity, then Achnter lists them as ${\displaystyle \aleph _{1},\aleph _{2},\ldots ,\aleph _{n}}$, omitting ${\displaystyle \aleph _{0}}$ for some reason and not clearly including ${\displaystyle \aleph _{\omega }}$ or greater, and perhaps more importantly for our context, not seeming to notice that this is not the same as the "set of all transfinite sets". Then later in the same section he refers to "the absolute infinity תּ" (look closely; that's a different letter), and later says that תּ is the "set of all sets". This is all on page 40.

I have to say I don't find this very convincing as an argument that we should refer to absolute infinity as either ת or תּ in the lead sentence. It might not be terrible to mention it somewhere in the body. --Trovatore (talk) 04:46, 10 December 2025 (UTC)

@Trovatore @MEN KISSING, I apologize for my delay, I haven't had much time to work on Wikipedia. I notice there is some discussion at Wikipedia:Redirects for discussion/Log/2025 December 25#Tav (number) too. I'm going to reply to as much as I can here to keep relevant discussion together.

To respond to the first couple things, Cantor noticed the Aleph paradox early in his development of set theory. This became his default example of a collection which is absolutely infinite and, in later writings and would even use Tav and "Absolute infinite" in places where the other would make more sense. This was about as far into the history as I got before I decided to "correct" the symbol since I couldn't find anything like that for Omega. I still believe that if there is a symbol for "Absolute infinite" then that symbol is Tav at least for historical reasons, however, after reading further, there are good reasons to not use it as such that I'll expand on later.

About Achtner, I agree that the written ${\displaystyle \aleph _{1},\aleph _{2}...\aleph _{n}}$ is problematic, but the whole section is informal and honestly, I wouldn't blame Achtner for the mistakes or confusing wording. Reading Cantor directly is difficult since he tends to be extremely hand-wavey, appealing to things like intension, intuition, theology, or nothing at all to define the concepts he's using. You really get a sense as to why so many mathematicians brushed off his ideas as nonsense early on. It wasn't until Russell, von Neumann, Zermelo, etc. managed to formalize his concepts in a more sound way that they could be explored by those who could not ask Cantor for clarification directly. Unfortunately, it seems Absolute Infinite did not get this treatment, and remains a very hand-wavey concept. Possibly because Cantor was very adamant that it was "not conceivable by mathematics" and pushed the theological aspect. But I digress. One last thing; about the Tav-dot symbol, it doesn't seem like Cantor ever used that notation, and it doesn't seem like Achtner is trying to imply that Cantor did. Introducing Tav as "He called this set [Tav], or absolute infinity...", and introducing Tav-dot as "Let [Tav-dot] be the set of all sets." (both page 40).

It seems Cantor initially took Absolute infinite as a kind of actual infinite early on. This is when he would conflate "absolute infinite" the size, and Tav the class of cardinals. Later, after deliberation with others, he would take the stance that it was a potential infinite, only being able to be approached, but no completed "absolute infinite" exists. Around here is when he stops conflating Tav and the Absolute. Since Tav never caught on as a notation, and Cantor himself ceased using it eventually, it does seem best to not identify it as the symbol for the absolute. All of this was before the formal use of proper classes, which would likely serve to reverse the potential-vs-actuality of the Absolute in a more modern analysis.

Personally, I think the best way forward for this article in the mathematical aspect is through the Axiom of limitation of size / the Limitation of size philosophy that was prominent at the time, and as such the best object to define the absolute would be the Universe of Sets, (or, more intentionally correct, the cardinality of that object) but that seems like a future discussion. – Farkle Griffen (talk) 19:51, 29 December 2025 (UTC)

The redirect Tav (number) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 December 8 § Tav (number) until a consensus is reached. Largoplazo (talk) 18:28, 8 December 2025 (UTC)

• @Farkle Griffen, Trovatore, and Prokurator11: Hello! I'm here from that RfD. Based on the lack of !votes on there besides mine, we have no idea what to do with it. Apparently it used to be a duplicate article that was merged here, but it's unclear whether Tav is actually a common symbol for absolute infinity as a number. I was referencing the discussion you folks had here, but there doesn't seem to be much of a conclusion to work with. Since it would be relevant for improving this article anyways, could you three talk about this a bit more? Is Tav in any sense notation for absolute infinite as a number?
  Thanks and have a lovely rest of the year! MEN KISSING (she/they) T - C - Email me! 04:14, 27 December 2025 (UTC)