Talk:Actual infinity

Improvements

The English could be improved a bit by a native speaker. 217.94.206.120 20:33, 30 March 2007 (UTC)Regards, WM

Probably there are English translations of some books quoted. 217.94.206.120 20:57, 30 March 2007 (UTC) Regards, WM

The whole page consists of pseudo-philosophical gibberish, and it should be deleted altogether. It is clearly conceived as a propaganda pamphlet by people who are completely devoid of any understanding of mathematics, but nonetheless for some reason have a serious issue with it. Let me stress: in mathematics, there is no such thing or object whatsoever as "actual infinity." For Wikipedia to say otherwise is an embarrassment and a disservice to the interested public.Kluto (talk) 09:55, 22 February 2013 (UTC)

Can there be infinitely many finite natural numbers?

IN-finite means NOT-finite. It is clear that the size of the set of naturals is greater than any "natural number", are finists implying there's something between infinite and finite? How they derived A2 from A is in my opinion erroneous.

Do constructivist share this view?Standard Oil (talk) 13:57, 27 April 2009 (UTC)

My math is sharper than my philosophy, so perhaps I'm misunderstanding the intent of this section. But let me plainly ask: are these philosophers' heads lodged in their rectums, or is their discussion actually more profound than gibberish spouted based on misunderstandings of set theory? --Dzhim (talk) 06:05, 28 May 2009 (UTC)

Yes I share your view that these "philosophers" have taken set theory too seriously. A formalist like me would view these infinities as statements in a formal language, so it's meaningless to ask how big an inaccessible cardinal really is (can anyone imagine it?). It even says in the article

Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking - a remarkable harmony between being and thinking. (D. Hilbert [6, 190])Standard Oil (talk) 14:09, 31 May 2009 (UTC)

Is this a real argument? It sounds suspiciously like something someone made up on the spot. I cannot think of a sense in which the axiom A implies the axiom A2. After all, for any natural number n, the sequence eventually gets larger than n, at the n+1th element. I'm tempted to delete this unless someone can provide citation that this argument actually is used by finitists. Or, at least, to replace A with A2, instead of leaving the strangest implication that A implies A2. As it is, I'm replacing the term "series" with "sequence", because "series" is used for sums. 124.120.128.36 (talk) 07:32, 4 June 2009 (UTC)

It's clearly a more philosophical argument than a logical one. After all finitists don't believe in infinity which is a consequence if we adhere to logic only. I recommend removing it so new comers won't get confused (like me 6month ago). Standard Oil (talk) 14:01, 5 June 2009 (UTC)

Is the Hilbert [6] not referenced properly? I'd like to read the original source but can't find it a reference to what the source actually is here.

Done. 124.122.141.37 (talk) 11:11, 6 June 2009 (UTC)

Bolding Good (for all articles on Wikipedia)

I like the bolding which can help an overwhelmed newcomer/novice to the page navigate the material. This bolding could improve all (many) articles on Wikipedia because it would give more levels of detail to peruse through (Large font titles first, bolded important sentences within titled sections, and then the text). Of course, I know the whole article is really supposed to be an "introduction" anyway. —Preceding unsigned comment added by 71.164.236.198 (talk) 21:29, 18 June 2009 (UTC)

References

I object to two of the references. Both of them are in German and of little use to the reader. The first is Mückenheim's book. Wolfgang Mückenheim (WM) has also edited this page and probably inserted the reference to his essentially self-published book. Anyone with a basic understanding of mathematics can examine what he has uploaded to the arxiv to see that his reasoning is not to be trusted. This reference had been removed before.

The website by Sponsel lists links without differentiating between scholarly sources and mere cranks. Also, he would not be qualified to make such a distinction. 130.149.15.196 (talk) 10:46, 3 March 2010 (UTC) (Carsten Schultz, TU Berlin)

If in your estimation that these sources were not used to write the article, and given they are not used for quotation purposes etc, then I suggest you go ahead and remove them. Bill Wvbailey (talk) 19:02, 3 March 2010 (UTC)

Now that's a tough one: Did Mückenheim use his own book when writing large parts of this article? 87.160.141.93 (talk) 10:10, 4 March 2010 (UTC) (CS again)

The references make no sense. For example, in 'A. Fraenkel [4, p. 6]' and 'D. Hilbert [6, p. 169]', what do the 4 and the 6 refer to? Someone needs to take this in hand. 86.132.223.248 (talk) 19:58, 12 February 2017 (UTC)

I'm hungry

pi anyone? 188.29.165.163 (talk) 12:48, 14 January 2015 (UTC)

Long list of quotations with no context

The final two sections look like someone's scrapbook on infinity. I don't think that the reader is served by such a long list of quotes without any context or explanation. Phiwum (talk) 13:27, 14 March 2016 (UTC)

The Leibniz quote

While the Valspeak translation is entertaining, it may be better to give it in its context: "Je suis tellement pour l'infini actuel [we currently stop here], qu'au lieu d'admettre que la nature l'abhorre, comme l'on dit vulgairement, je tiens qu'elle l'affecte partout, pour mieux marquer les perfections de son auteur." Double sharp (talk) 13:20, 23 May 2016 (UTC)

Pre-Socratic?

This article weirdly starts with "Pre-Socratic" then jumps to "Aristotle". If Socrates isn't interesting, why doesn't it start with "Pre-Aristolean"? William M. Connolley (talk) 20:25, 20 December 2017 (UTC)

Can't be a serious question: Compare search results for "Pre-Socratic" and "Pre-Aristolean". Pre-Socratic philosophy is a well established term for a period in the history of western philosophy. One wonders, what the knowledge of WP editors might be...--2.247.255.159 (talk) 04:13, 21 December 2017 (UTC)

The first sentence makes no sense

The first sentence: "Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set", makes no sense as any kind of definition of actual infinity. Paul August ☎ 18:28, 27 December 2017 (UTC)

Re-reading it (don't look at me, I didn't write it) I think the first sentence just isn't needed or helpful; so I tried removing it William M. Connolley (talk) 00:22, 28 December 2017 (UTC)

Yes, much better. Paul August ☎ 00:49, 28 December 2017 (UTC)

Examples in Lead

The examples given seem to contradict each other. What could the distinction between "adding one to each number before it" and the natural numbers be? The cited source isn't useful relevant to the article either, it only discusses infinity, as opposed to "Actual infinity". I believe that line should be deleted, but do not want to be overly hasty. Tototavros (talk) 00:20, 4 February 2020 (UTC)

"Adding one to each number before it" is a process. The natural numbers are a set. They're quite different animals. Aristotle (and Gauss) acknowleged that one could keep adding one but objected to the idea that the process could somehow be completed, thus producing an actual infinite entity.

As regards citations, the first does use the phrase "actual infinity" while the second prefers "completed infinity", meaning the same thing.

Peter Brown (talk) 01:58, 4 February 2020 (UTC)

Missing defense of actual infinity!

The article, as currently written, does not seem to offer any sharp description of actual infinity, or defend why anyone might ever believe in it. It seems mostly a compendium of quotes to reject it. I would describe the success of actual infinity in mathematics in this way (my take may be non-standard, but seems to be what actually happens):

Actual infinity is now commonly accepted, because mathematicians have learned how to construct algebraic statements using it. For example, one may write down a symbol, $\omega$, with the verbal description that "$\omega$ stands for completed (countable) infinity". This symbol may be added as an ur-element to any set. One may also provide axioms that define addition, multiplication and inequality; specifically, ordinal arithmetic, such that expressions like $n<\omega$ can be interpreted as "any natural number is less than completed infinity", or even statements such as $\omega <\omega +1$ are possible and consistent. The theory is so well developed, that even rather complex algebraic expressions, such as $\omega ^{2}$, $\omega ^{\omega }$ and even $2^{\omega }$ can all be given a precise verbal description, and can be used in a wide variety of theorems and claims which appear to be consistent and meaningful. This detailed richness, the ability to define ordinal numbers in a consistent, meaningful way, renders much of the debate moot: whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with actual infinities using the tools of algebra is clearly in hand.

I'm gonna be bold and slot this in somewhere, and perhaps someone can improve upon this. 67.198.37.16 (talk) 23:45, 5 June 2021 (UTC)

I disagree that the formal theory of ordinal numbers is relevant to any philosophical question about how infinity works or whether it exists. The theory does not require any "actual infinities" to exist. 47.35.146.128 (talk) 14:49, 1 July 2022 (UTC)

Cantor

Cantor was the inventor of transfinite, the first mathematician to distinguish two types of actual infintie and to clearly demonstrate the Absolute can't be described by numbers and Maths.

This is a basic result for Mathematical science. In philosophy, the same result was given by Aristotle some centuries before. Zeno of Elea had just demonstrated that an actual infinite can't move by itself nor by other moving bodies. Aristotle conceived the Supreme Being as the immobile mover. Aristotle demonstrated that an actual infinite can't exist in the space and time of physics, where any quantity can be solely defined as continous, to say, infinitely divisible, whereas the actual infnite shall be unique and not divisible.

This statement didn't exclude the existence of an actual infinite above the world of number. The existence of God, conceived as an actual and trascendent infinite, was at least made known by the works of St. Thomas Aquinas.

78.14.138.162 (talk) 11 June 2021

Many problems with this.

• The transfinite was not invented by Cantor. Space had infinitely many points well before Cantor was born.
• The Absolute is not a mathematical concept, so its properties are not a basic mathematical result.
• Zeno, who died around 15 years before Aristotle was born, had not "just" demonstrated anything when Aristotle presented his views.
• Whether space or time is infinite in extent is unknown. Aristotle did not "demonstrate" that they are not infinite.
• Continuity is not the same as infinite divisibility.
• Atheism is a viable theory. Thomas Aquinas did not "make known" God's existence.

Peter Brown (talk) 16:59, 12 June 2021 (UTC)

Doubts about the new short description

John Maynard Friedman has added "Concept in the philosophy of mathematics" as the short description for the article. Prior to the 19th century, yes, discussion of actual infinity was confined to philosophers, and Cantor's defense of the notion necessarily involved philosophical argumentation. As noted in the section § Current mathematical practice, however, "Actual infinity is now commonly accepted," so that discussion has turned to the mathematics of actual infinity. While intuitionism and other finitistic approaches have not quite disappeared, a modern presentation to a general audience has no need defend statements like "$\mathbb {Z}$ [the set of integers] is a subset of the set of all rational numbers $\mathbb {Q}$", found in the Integer article; the terms of the statement are, today, nearly uncontroversial mathematical concepts. This no longer lies in the domain of philosophy. A better short description, perhaps, would be "Concept in mathematics". That would still be misleading, though, as much of the article is devoted to pre-19th-century thought. Ideas? Peter Brown (talk) 20:35, 4 November 2021 (UTC)

@Peter M. Brown: Sometimes my adding a wp:short description to an article provokes a more knowledgeable editor into improving it. So absolutely go ahead and change it. This article didn't have an SD, so I gave it one – essentially by filleting the opening sentence. The actual mathematics involved is way above my pay grade.

The party line on short descriptions is that they are for the Wikipedia app, so that when visitors search for an article using the word "infinity", they are presented with a list of probably relevant articles and their associated short descriptions. Its espoused purpose is to distinguish, not to inform, so 40 characters is adequate. BUT they are also very useful in turning a See Also list of often cryptic (to non-cognoscenti) article names into something useful, when the template {{annotated link}} is used – as I have been doing whenever I consider it appropriate. (See for example Tragedy of the commons#See also.)

Full disclosure: I am a strong proponent of the value of Wikipedia for discovery of ideas, that it is one of its shining success stories. We should do all we can to encourage serendipity. --John Maynard Friedman (talk) 23:53, 4 November 2021 (UTC)

Helpful background, thanks. When I search for "Infinity" using the app, however, the only articles suggested are those in which the initial word is either "Infinity" or "Infinite". With "Actual", I don't get Actual infinity but I do get Actual Fucking, an album by Cex. I am not inspired to improve on your SD when it only appears when one enters "Actual infinity" and then only Actual infinity is suggested.

Peter Brown (talk) 01:41, 5 November 2021 (UTC)

That's a misinterpretation of the relationship between philosophy of mathematics and mathematics itself. As I mentioned above, I don't think the section on current mathematical practice is actually relevant to the philosophical arguments regarding infinity (which is not necessarily to say that it's totally irrelevant to the article). The philosophical question was never "settled" and the reason actual infinity is accepted by most mathematicians is that working mathematicians are by and large not philosophers and don't worry about issues like this. Actual infinity is not a math concept, it's a philosophical concept. 47.35.146.128 (talk) 14:57, 1 July 2022 (UTC)

The axiom of Euclidean finiteness?

What actually is this "axiom of Euclidean finiteness" that states that actualities, singly and in aggregates, are necessarily finite? William M. Connolley (talk) 14:25, 17 May 2023 (UTC)

It DOES need sourced but is simply the axiom of the finitist camp.

Note: the other editors won't fix it because they can't find it in the article. Please not location in page. Victor Kosko (talk) 16:18, 18 May 2023 (UTC)

Unclarities and errors of the current lead paragraph

Coming from references from Zeno's paradoxes in various places around the wiki, esp. geometric series and related articles, I encountered this page and I find its lead paragraph confusing, incorrect, and daunting to correct. I'll try stating my confusions here and encourage more established editors to take a crack at revision; if there isn't a revision in short order then I'll try an edit of my own in no less than several days.

>In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an infinite series, infinite product, or limit.

(1) The parenthetical "(if the axiom of infinity is included)" seems to be a non-sequitur. Historically, it's certainly not essential since the considerations of potential and actual infinity arose in Anaximander, Plato, and Aristotle. I'd find it natural to simply delete this parenthetical or move it to a clause in the final sentence of the paragraph.

(2) Potential infinity as "non-terminating process" is specifically sequential and unidimensional here, whereas in Aristotle (just the case I know best off the cuff, not one I'd call authoritative) the distinction arises for the multidimensional spatial continuum as well (see usual places in the Physics and On Generation and Corruption; if I edit I'll find and cite page numbers), so this definition is incorrectly overspecific and must be edited to something more correct (and ideally with citations).

(3) The first examples are strangely chosen. Why "natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers"? Historically, the natural numbers and continua are the first key examples and I see a case for the transfinites; extended real numbers aren't a better example than the real open interval (0,1) and would be a confusing jump straight from the naturals to a set mixing of types of infinity (an infinity of one kind (infinite set cardinality) that further includes infinities of a different kind (infinite value) as parts). "Even an" seems like a style/tone error. And this example list does not include any infinite values except within sets of infinite cardinalities.

(4) "A non-terminating process produces" is crucially confusing: the philosophical debate often hinges exactly on whether the process does produce or just implies possibilities of production. At very least, some change to a subjunctive "would produce" is necessary here.

I'll stop there, and I hope that this gets some attention but if not I'll try a bold edit. I'm not familiar with the rest of Wikipedia's material on infinity, so I certainly won't start editing until I've reviewed that.

RowanElder (talk) 16:01, 10 September 2024 (UTC)

I agree that this lead was very poor. I have edited it for distinguishing the general philosophical question and its mathematical counterpart.

I am not competent for expanding the non-mathematical aspects. However, it is important to not confuse them with the mathematical concept, since the existence of mathematical "actual infinities" is no more a question in mathematics. D.Lazard (talk) 18:23, 10 September 2024 (UTC)

This is much improved, thank you. I agree it is important not to confuse any of the non-mathematical philosophical concepts with the contemporary mathematical concept, and I am solely concerned to ensure that the often-confused but encyclopedically-important historical mathematical concepts will be included accurately as well. (2) is the only one of the concerns above that was not addressed well by the edit and I should be able to take care of that myself, now, in a way that won't be controversial. Thanks again. RowanElder (talk) 21:49, 10 September 2024 (UTC)

I don't like the changes your making to the article. I'm your only decent anti-set-theorist.

The highest most professional set theorists claim size is not cardinality or originality, which is evident since size is precisely defined in set theory and infinite sets can have different relative size, cardinality, originality

Although cardinality pretends to be size it is not, it is size plus in the (it is somewhat long) definition of cardinal and cardinality various lies, so only your technique for comparing sets can be used but not a particular tree technique that proves the set of traps from 0 to 1 is the same size as the set of naturals, this technique can be used for ≤ or ≥ but not < or >, and if 2 infinite sets cannot be proven the same size they are different cardinality without proof, such that using the diagonal argument to prove lack of proof of =, you thus have proof per the definition of ≠ cardinality.

You thus giving in to appearances and saying cardinality is size is contradicting the top set theorists.

You should rather claim is not.

Wikipedia itself claims 2 sets can have different cardinal vs ordinal size change using the same exponential function, or likewise using the cardinal vs ordinal exponentiation, being you claim a set can have an ordinal and a cardinal size Victor Kosko (talk) 22:56, 12 September 2024 (UTC)

I'm sorry, but I don't understand this comment. Between the typos like originality/ordinality and the broken grammar, I simply can't parse this. Could you restate your point? Certainly size, ordinality, and cardinality are important to distinguish conceptually, but I don't see the relevance to my specific concerns above except that I used the word "cardinality." RowanElder (talk) 01:23, 13 September 2024 (UTC)

at 10:17 on 8 September D. Lazard edited the article with:

'A great discovery of Cantor is that, if one accept infinite sets, then there different sizes (cardinalities) of infinite sets, and, in particular, the cardinal of the continuum of the real numbers is strictly larger than the cardinal of the natural numbers.'

Again size is not at all Cardinality Victor Kosko (talk) 02:41, 14 September 2024 (UTC)

Alright, I see. I'd argue "not at all" is an overstatement; cardinality is clearly "a size" in the generic sense though few sizes are exactly the cardinality, like any metric defines "a distance" though few metrics are exactly Euclidean distances. And then D. Lazard's edit makes perfect sense to me when the parenthetical is understood as a technical specification that from within the generic class of informal size concepts "size" should mean formal "cardinality" here.

In any case this detail does not seem like a priority to me when there are much less contentious, less nuanced issues to solve, but in time hopefully the page will be clear on this point. I see you're passionate about the issue, but I'm not and I'm unlikely to become passionate about it now by persuasion here. RowanElder (talk) 04:16, 14 September 2024 (UTC)

you obviously didn't read my post

Cardinality is size plus

Your technique for comparing size

Cannot be used for > or < relative infinite size between sets

And

Using the diagonal argument to prove your technique for proving infinite sizes equal, won't thus prove same size for the 2 sets in question, is interpreted as proving not equal cardinality, even though not equal size was not proven only impossibility of proving equal infinite size

And I remind you it's not only my opinion size is not cardinality it's the professional set theorists.

If you saw my tree proof you'd back off Victor Kosko (talk) 04:31, 14 September 2024 (UTC)

"Back off" from editing the page? There have been no attacks here.

I did read these posts here and a small chunk of your contrib history and talk page. I'm simply not persuaded to share your framing of the issue or check the detail of your technical arguments. I truly don't care about this point like you do at this time and Wikipedia has clear policies that I'm not required to engage original research. Perhaps I'm not persuaded due to heuristics that I'll regret later, in retrospect, but that's life. I don't properly understand Skolem's paradox yet either, though my colleagues tell me that it's quite foundationally important when it comes to these size issues. There are many libraries-worths of important books that I'll never read, and your proof isn't remotely close to a priority for me. RowanElder (talk) 05:12, 14 September 2024 (UTC)