Talk:0.999...

Yet another anon

Moved to Arguments subpage

Image

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

There is no consensus to remove the image, and a rough consensus to keep it. Mr. Swordfish (talk) 21:42, 10 January 2025 (UTC)

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The image included at the top of this article is confusing. Some readers may interpret the image to mean that 0.999... represents a sequence of digits that grows over time as nines are added, and never stops growing. To make this article less confusing I suggest that we explicitly state that 0.999... is not used in that sense, and remove the image. Kevincook13 (talk) 17:31, 1 January 2025 (UTC)

I do not see how this is confusing. The caption reads: "Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely" - nothing remotely like "sequence... that grows over time". I cannot see how one could meaningfully add a comment that "0.999..." is not used in a sense that has not even been mentioned. Of course lots of people are confused: that is the reason for the article, which in an ideal world would not be needed. Imaginatorium (talk) 04:29, 2 January 2025 (UTC)

If a sequence of digits grows over time as nines are added, and never stops growing, it is reasonable to conclude that the digit nine is repeating infinitely. Kevincook13 (talk) 18:14, 2 January 2025 (UTC)

Yes, notation 0.999... means that the digit nine is repeating infinitely. So, the figure and its caption reflect accurately the content of the article. D.Lazard (talk) 18:28, 2 January 2025 (UTC)

When we use the word repeating we should expect that some people will think we are referring to a process which occurs over time, like the operation of a Repeating firearm. Kevincook13 (talk) 22:03, 2 January 2025 (UTC)

You can think of this as a "process" if you like. 0.9999... means the limit of the sequence [0.9, 0.99, 0.999, 0.9999, ...]. Of course in mathematics nothing ever really "occurs over time", though I suppose you could consider it a kind of algorithm which if implemented on an idealization of a physical computer with infinite memory capacity might indefinitely produce nearer and nearer approximations. –jacobolus (t) 22:20, 2 January 2025 (UTC)

I think you are going in a very productive direction. We should explain to readers how what they might think we mean, "occurring over time", relates to what we actually mean. Kevincook13 (talk) 00:43, 3 January 2025 (UTC)

I personally think that would be distracting and not particularly helpful in the lead section. There is further discussion of this in § Infinite series and sequences, though perhaps it could be made more accessible. –jacobolus (t) 03:42, 3 January 2025 (UTC)

Yes, I agree that detailed discussion does not belong in the lead section. I personally think that the image is distracting and not helpful. In the lead section we can simply state that in mathematics the term 0.999... is used to denote the number one. We can use the rest of the article to explain why. Kevincook13 (talk) 16:23, 3 January 2025 (UTC)

Except that it's not true that 0.999... denotes the number one. It denotes the least number greater than every element of the sequence 0.9, 0.99, 0.999,... It's then a theorem that the number denoted in this way is equal to one. Tito Omburo (talk) 16:31, 3 January 2025 (UTC)

It also denotes the least number greater than every number which is less than one, just as 0.333...denotes the least number greater than every number which is less than one-third. That's why we say it denotes 1/3, and why we also say that the one with 9s denotes 1. Imaginatorium (talk) 17:39, 3 January 2025 (UTC)

@Tito Omburo, notice that @Imaginatorium just wrote above "we also say that the one with 9's denotes 1". The description "the least number greater than every element of the sequence 0.9, 0.99, 0.999,..." does describe the number one, just as does "the integer greater than zero and less than two". Kevincook13 (talk) 18:21, 3 January 2025 (UTC)

This is an incorrect use of the word "denotes". Denotes an equality by definition, whereas one instead has that 0.999... and 1 are judgementally equal. For example, does "All zeros of the Riemann zeta function inside the critical strip have real part 1/2" denote True or False? Tito Omburo (talk) 18:56, 3 January 2025 (UTC)

I think you are inventing this - please find reliable sources (dictionaries and things) to back up your claimed meaning of "denote". Imaginatorium (talk) 04:55, 9 January 2025 (UTC)

I agree that it is better to write that the term is used to denote the number one, rather than that the term denotes the number one. Kevincook13 (talk) 20:06, 3 January 2025 (UTC)

Its not "used to denote". It is a mathematical theorem that the two terms are equal. Tito Omburo (talk) 20:46, 3 January 2025 (UTC)

I think we can make this issue very clear. Assume that x equals the least number greater than every element of the sequence 0.9, 0.99, 0.999,... . Applying the theorem we learn that x = 1. Substituting 1 for x in the opening sentence of this article we have: In mathematics 0.999... denotes 1. If we also insist that 0.999... does not denote 1, we have a contradiction. Kevincook13 (talk) 18:45, 4 January 2025 (UTC)

You have redefined the word "denote" to mean precisely the same as "is equal to", which is confusing and unnecessary. It's better to just say "is equal to" when that's what you mean, so that readers are not confused. –jacobolus (t) 18:56, 4 January 2025 (UTC)

I agree that redefining the word denote would be confusing and unnecessary. I simply defined a variable x to be equal to a number, the least number. Kevincook13 (talk) 20:04, 4 January 2025 (UTC)

I'm in agreement with @Imaginatorium and @D.Lazard on this. The image does not suggest a process extended over time, and it correctly reflects the (correct) content of the article, so there is no need to remove it. I'm not persuaded that people will interpret "repeating" as purely temporal rather than spatial. If I say my wallpaper has a repeating pattern, does this confuse people who expect the wallpaper to be a process extended over time? (Are there people who think purely in firearm metaphors?) MartinPoulter (talk) 17:30, 3 January 2025 (UTC)

Consider the number 999. Like the wallpaper, it contains a repeating pattern. That pattern could be defined over time, one nine at a time. Or it could be defined at one time, using three nines. Kevincook13 (talk) 18:27, 3 January 2025 (UTC)

Is it OK if I go ahead and edit the article, keeping in mind all the concerns which have been raised with my proposed changes? Kevincook13 (talk) 17:56, 8 January 2025 (UTC)

Can you be more specific about which changes you want to implement? MartinPoulter (talk) 20:32, 8 January 2025 (UTC)

The first change would be to remove the image. Kevincook13 (talk) 15:06, 9 January 2025 (UTC)

I'm confused, @Kevincook13. Where in the above discussion do you see a consensus to remove the image? You have twice said the image should be removed, and I have said it should stay. No matter how many times you express it, your opinion only counts once. Other users have addressed other aspects of your proposal. Do you sincerely think the discussion has come to a decision about the image? MartinPoulter (talk) 13:47, 10 January 2025 (UTC)

No. I do not think there is agreement on removing the image. (I don't personally think it is spectacularly good, but the argument for removing it appears to me to be completely bogus.) Imaginatorium (talk) 04:57, 9 January 2025 (UTC)

The term 0.999... is literally a sequence of eight characters, just as y3.p05&9 is. Yet, the term itself implies meaning. I think confusion about the term can be reduced simply by acknowledging different meanings the term might imply. It does imply different meanings to different people. We can respect everyone, including children who are not willing to simply accept everything a teacher tells them. We can do our best to help everyone understand what we mean when we use the term. Kevincook13 (talk) 15:32, 9 January 2025 (UTC)

For example, if a child thinks that by 0.999... we mean a sequence of digits growing over time, and the child objects when told that the sequence of digits is equal to one, we can respond by saying something like the following: You are correct that a growing sequence of digits does not represent one, or any number, because the sequence is changing. We don't mean that 0.999... represents a changing or growing sequence of digits. Kevincook13 (talk) 16:12, 9 January 2025 (UTC)

We don't mean a changing or growing sequence of digits. That is what it is confusing to say that we mean a repeating sequence of digits. Kevincook13 (talk) 16:15, 9 January 2025 (UTC)

What we mean is a number. Kevincook13 (talk) 16:18, 9 January 2025 (UTC)

This article is about the meaning of 0.999... in mathematics not about the possible meanings that people may imagine. If people imagine another meaning, they have to read the article and to understand it (this may need some work), and they will see that their alleged meaning is not what is commonly meant. If a child objects to 0.999... = 1, it must be told to read the elementary proof given in the article and to say which part of the proof seems wrong. D.Lazard (talk) 16:58, 9 January 2025 (UTC)

What do we mean by the term number? A number is a measure, not a sequence of digits. We may denote a number using a sequence of digits, but we don't always. Sometimes we denote a number using a word, like one. Sometimes we use a phrase such as: the least number greater than any number in a certain sequence. We may use a lowercase Greek letter, or even notches in a bone. Kevincook13 (talk) 16:44, 9 January 2025 (UTC)

By the term "number", we mean a number (the word is not the thing). It is difficult to define a number, and this took several thousands years to mathematicians to find an acceptable definition. A number is certainly not a measure, since a measure requires a measurement unit and numbers are not associated with any measurement unit. The best that can be said at elementary level is something like "the natural number three is the common property of the nines in 0.999..., of the consecutive dots in the same notation, and of the letters of the word one". D.Lazard (talk) 17:20, 9 January 2025 (UTC)

I see. A number is not a measure, but it is used to measure. Thanks. Kevincook13 (talk) 17:40, 9 January 2025 (UTC)

A number is a value used to measure. Kevincook13 (talk) 17:42, 9 January 2025 (UTC)

The caption on the image is: Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely.

The caption can be understood to mean that the term 0.999... is a zero followed by a decimal point followed by the digit 9 repeating infinitely, which meaning is distinct from the meaning that 0.999... denotes the number one.

If we retain the caption, we may communicate to readers that we mean that 0.999... is a repeating sequence, which sequence denotes the number one. That doesn't work because repeating sequences themselves cannot be written completely and and therefore cannot be used to notate.

0.999... is notation. The purpose of this article should be to help others understand what it denotes. If it denotes a repeating sequence of digits, then we should say so in the lead sentence. Kevincook13 (talk) 18:32, 9 January 2025 (UTC)

How does the first sentence of the article not explain that notation? The meaning of the notation is the smallest number greater than every element of the sequence (0.9,0.99,...). Tito Omburo (talk) 18:39, 9 January 2025 (UTC)

Because it does not make sense to say that the sequence is repeating, because all the nines have not already been added, and at the same time to say that the sequence represents a number, because all the nines have already been added. It is confusing because it is contradictory.

When we say that the sequence is repeating, people who are not trained in mathematics will likely assume that we mean that all the nines have not already been added, and therefore that the sequence is changing and therefore, does not represent a number. Which, I believe, is why the subject of this article is not more widely understood. Kevincook13 (talk) 19:05, 9 January 2025 (UTC)

I think I understand part of the confusion, which I've hopefully tried to correct with an edit. The notation 0.999... refers to a repeating decimal, a concept which had not been linked. There is a way of associating to any decimal expansion a number as its value. For the repeating decimal 0.999..., that number is 1. Tito Omburo (talk) 19:09, 9 January 2025 (UTC)

I like the edits. Because the least number is one, the meaning of the lead sentence can be understood to be that 0.999... is a recurring decimal whose value is defined as one. The notation below should match. Instead of $0.999...=1$, we should write $0.999...\ {\overset {\underset {\mathrm {def} }{}}{=}}\ 1$. Kevincook13 (talk) 19:40, 9 January 2025 (UTC)

No. The point is that the notation has a definition which is a standard one for repeating decimals of this form. It is a theorem that this number is one, but that is not the definition. Tito Omburo (talk) 19:47, 9 January 2025 (UTC)

I agree. What you are saying agrees with what I am saying. It is a theorem that the least number is one, not a definition. The notation has a standard definition which defines the notation to be equal to the least number, whatever that least number is.

1. Given that the notation is defined to be equal to the least number
2. And given a theorem that the least number does equals one
3. Therefore the notation is defined to be equal to a number which does equal one.
4. Note that it does not follow from the givens that the notation is equal to one, or that the notation is equal to the least number.

Kevincook13 (talk) 20:23, 9 January 2025 (UTC)

This is not correct, but I feel like we're talking in circles here. Cf. WP:LISTEN.

Let me try one more thing though. If we wanted a more explicit definition of 0.999..., we might use mathematical notation and write something like $$0.999\ldots \ {\stackrel {\text{def}}{=}}\ \sum _{k=1}^{\infty }9\cdot 10^{-k}=1.$$ This is discussed in the article in § Infinite series and sequences. –jacobolus (t) 02:58, 10 January 2025 (UTC)

Can you see that the summation is a process which must occur over time, and can never end? Do you notice that k cannot equal 1 and 2 at the same time? However, if we insist that the summation does occur all at once, then we affirm that k does equal 1 and 2 at the same time. We affirm that we do intend contradiction. If so, then we should clearly communicate that intention. Kevincook13 (talk) 15:14, 10 January 2025 (UTC)

Please stop misusing the word denotes when you mean "is equal to". It's incredibly confusing. –jacobolus (t) 20:57, 9 January 2025 (UTC)

I agree that the difference between the two is critical. I've tried to be very careful. Kevincook13 (talk) 21:13, 9 January 2025 (UTC)

I don't know if this will help at all, but it may. I think that we have been preoccupied with what infinity means, and have almost completely ignored what it means to be finite. We don't even have an article dedicated to the subject. So, I have begun drafting one: Draft:Finiteness. Kevincook13 (talk) 00:00, 10 January 2025 (UTC)

I think the problem here is that there are two levels of symbol/interpretation. The literal 8-byte string "0.999..." is a "symbol for a symbol", namely for the infinitely long string starting with 0 and a point and followed by infinitely many 9s. Then that infinitely long symbol, in turn, denotes the real number 1.

It's also possible that people are using "denote" differently; I had trouble following that part of the discussion. But we need to be clear first of all that when we say "0.999..." we're not usually really talking about the 8-byte string, but about the infinitely long string. --Trovatore (talk) 05:06, 10 January 2025 (UTC)

This is also a weird use of "denote", in my opinion. For me, the word denote has to do with notation, as in, symbols that can be physically written down or maybe typed into a little text box. For example, the symbol ⁠$\pi$⁠ denotes the circle constant. The symbol ⁠$1$⁠ denotes the number one. The mathematical expression ⁠$ax^{2}+bx+c=0$⁠ denotes the general quadratic equation with unknown coefficients.

An "infinitely long string" is an abstract concept, not anything physically realizable, not notation at all. From my point of view it doesn't even "exist" except as an idea in people's minds, and in my opinion it can't "denote" anything. But again, within some abstract systems this conceptual idea can be said to equal the number 1. –jacobolus (t) 07:05, 10 January 2025 (UTC)

While you can't physically use infinitely long notation, I don't see why it should be thought of as "not notation at all". Heck, this is what infinitary logic is all about. In my opinion this is the clearest way of thinking about the topic of this article — it's an infinitely long numeral, which denotes a numerical value, which happens to be the real number 1. The reason I keep writing "the real number 1" is that this is arguably a distinct object from the natural number 1, but that's a fruitless argument for another day. --Trovatore (talk) 07:15, 10 January 2025 (UTC)

The infinitely long string. The one that is not growing over time because it already has all of the nines in it, and because it is not growing can be interpreted as a number. The one that is repeating, because it does not at any specific instance in time have all the nines yet. That one? The one that is by definition a contradiction? Kevincook13 (talk) 15:56, 10 January 2025 (UTC)

"By definition a contradiction". Huh? What are you talking about? If you can find a contradiction in the notion of completed infinity, you're wasting your time editing Wikipedia. Go get famous. --Trovatore (talk) 18:25, 10 January 2025 (UTC)

Above, I just described P and not P, a contradiction. Kevincook13 (talk) 19:26, 10 January 2025 (UTC)

Um. No. You didn't. I would explain why but in my experience this sort of discussion is not productive. You're wandering dangerously close to the sorts of arguments we move to the Arguments page. --Trovatore (talk) 21:13, 10 January 2025 (UTC)

I'm not wasting my time. I believe in Wikipedia. Kevincook13 (talk) 19:33, 10 January 2025 (UTC)

We look to famous people to tell us what to understand? Kevincook13 (talk) 19:40, 10 January 2025 (UTC)

I see Wikipedia as a great place for people to learn about and evaluate the ideas of people who, over time, have become famous for their ideas. Kevincook13 (talk) 20:04, 10 January 2025 (UTC)

The fact of the matter is that if any theory logically entails a contradiction, then that theory is logically inconsistent. If we accept logical inconsistency as fact, then we can save everyone a lot of time by saying so. Kevincook13 (talk) 20:19, 10 January 2025 (UTC)

I suggest that we address each of the following in our article:

1. The 8-byte term
2. (0.9, 0.99, 0.999, ...)
3. The least number
4. The growing sequence
5. The contradiction

Kevincook13 (talk) 17:11, 10 January 2025 (UTC)

There is no contradiction. There is no growing sequence. 0.999... is indeed infinitely long, and = 1. Hawkeye7 (discuss) 21:14, 10 January 2025 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

"0.999999999999999999999999999999999999999" listed at Redirects for discussion

The redirect 0.999999999999999999999999999999999999999 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 March 18 § 0.99999.... until a consensus is reached. Rusalkii (talk) 03:47, 18 March 2025 (UTC)

Possible redirects of similar nature

Below shows a list of all possible redirects of similar nature.

• 0.9 (not a redirect)
• 0.99 (not a redirect)
• 0.999
• 0.9999
• 0.99999
• 0.999999
• 0.9999999
• 0.99999999
• 0.999999999
• 0.9999999999
• 0.99999999999
• 0.999999999999
• 0.9999999999999
• 0.99999999999999
• 0.999999999999999
• 0.9999999999999999
• 0.99999999999999999
• 0.999999999999999999
• 0.9999999999999999999
• 0.99999999999999999999
• 0.999999999999999999999
• 0.9999999999999999999999
• 0.99999999999999999999999
• 0.999999999999999999999999
• 0.9999999999999999999999999
• 0.99999999999999999999999999
• 0.999999999999999999999999999
• 0.9999999999999999999999999999
• 0.99999999999999999999999999999
• 0.999999999999999999999999999999
• 0.9999999999999999999999999999999
• 0.99999999999999999999999999999999
• 0.999999999999999999999999999999999
• 0.9999999999999999999999999999999999
• 0.99999999999999999999999999999999999
• 0.999999999999999999999999999999999999
• 0.9999999999999999999999999999999999999
• 0.99999999999999999999999999999999999999
• 0.999999999999999999999999999999999999999
• 0.9999999999999999999999999999999999999999
• 0.99999999999999999999999999999999999999999
• 0.999999999999999999999999999999999999999999
• etc. (all such links with 42 or more 9s are red)

GTrang (talk) 02:44, 24 April 2025 (UTC)

That was satisfying to scroll through. Ca ^{talk to me!} 03:39, 23 May 2025 (UTC)

I removed the excessive list (200+ entries) of red links. –jacobolus (t) 04:21, 23 May 2025 (UTC)

Great, now I feel robbed of the satisfying scrolling Ca got to experience... —Locke Cole • t • c 18:32, 2 June 2025 (UTC)

Knock yourself out: special:permalink/1291742170#Possible redirects of similar nature. –jacobolus (t) 23:31, 2 June 2025 (UTC)

Why does this have so many redirects? Lxvgu5petXUJZmqXsVUn2FV8aZyqwKnO (talk) 04:03, 21 June 2025 (UTC)

They don't serve much purpose, but they also don't hurt anything. As they say, "redirects are cheap". –jacobolus (t) 20:34, 21 June 2025 (UTC)

Surreal number system

0.999... is not equal to 1 in the surreal number system. This article is spreading misinformation.

https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-to-1-surreally-this-is-not-so/ https://mindyourdecisions.com/blog/2014/09/17/surreal-0-999-repeating-is-not-equal-to-1-video/ 172.13.197.175 (talk) 18:11, 2 June 2025 (UTC)

The article mentions the surreal number system once and does not claim that they are equal. EvergreenFir (talk) 18:14, 2 June 2025 (UTC)

A search of the archives turns up a number of hits, but this one seems to have directly addressed it: Talk:0.999.../Arguments/Archive_3#Reals_vs._Surreals. —Locke Cole • t • c 18:30, 2 June 2025 (UTC)

The bulk of this article is about the real numbers, or at least that's my reading of it and I would imagine that most of our readers are making that assumption too. There's a later section on alternative_number_systems, which contains the one brief mention of the surreal numbers but the rest of the article is about the reals (Although it also applies to subsets such as the rationals and and supersets such as the complex numbers, discussing that would probably just be a distraction for most readers)

That said, I don't think it hurts anything to make it clear at the outset that we're talking about the reals and not some other system, so I'll implement that minor change. Mr. Swordfish (talk) 23:51, 21 June 2025 (UTC)

Is the Archimedean property an axiom of the real numbers?

The current version of the article contains the following phrase:

...the Archimedean property, a defining axiom of the real number system

I am unaware of any source that states the Archimedean property as an axiom of the real numbers. Conventionally, the axiom expressing Completeness of the real numbers is either Dedekind completeness or the Least-upper-bound property, although there are other equivalent statements. While the Archimedean property follows from any of these axioms, I don't think that it is equivalent and therefore could be substituted as an axiom.

If not, then we need to correct the article. If so, then we'd need a source and we're still faced with the issue that the article is at variance with the "usual" literature. Or at least with the ones I'm familiar with.

I'm correcting the language pending further discussion here. Mr. Swordfish (talk) 23:43, 21 June 2025 (UTC)

It is sometimes described as an axiom (the "axiom of Archimedes"), and I think axiom works better in this context than property. E.g., it is one of Hilbert's axioms. Tito Omburo (talk) 11:55, 22 June 2025 (UTC)

Many things are called "axioms" that are not a "defining axiom of the real numbers". This appears to be an example since although the Archimedean property can be proved from the Dedekind completeness or the Least-upper-bound property axioms, those do not follow from the Archimedean property. i.e. The Archimedean property is not strong enough to serve as an axiom of the real numbers. It's not, so let's not say that it is. Mr. Swordfish (talk) 12:59, 22 June 2025 (UTC)

I mean, by itself completeness is also "not strong enough to serve as an axiom of the real number system", viewed in isolation. Completeness axiom + ordered axiom. Or completion of rationals + Archimedean axiom. Both give you the real numbers. (The latter was Hilbert's viewpoint. And if we're worried about independence of the axioms defining the rals, something rarely discussed, here is a list of axioms, proven independent, which includes a property equivalent to the Archimedean property.) But my point is that the Archimedean property is an axiomatic property, in the colloquial sense: something assumed to be true of the structure. If some other set of axioms didnt imply the Archimedean property, we wouldn't accept them as defining a "real number". It was foundational, and preceded all other attempts to systematize the real number system. Tito Omburo (talk) 16:16, 22 June 2025 (UTC)

The usual axiomatic treatment of the reals is to list the field axioms, the order axioms that make it an ordered field, and some version of completeness. The three together characterize the reals up to isomorphism.

If there's a source that develops real analysis from the Archimedian property as an axiom we might be technically correct calling it an an "axiom of the real numbers", but we'd be promoting a viewpoint that sufficiently unusual that it is possibly WP:FRINGE. Mr. Swordfish (talk) 16:39, 22 June 2025 (UTC)

I would encourage you to retract your viewpoint that this is WP:FRINGE. That is simply not tenable. It was one of the first axioms in history! (And the "usual axiomatic treatment" is a bit of a lie, isn’t it. The "usual axioms" that appear in many real analysis texts are not independent! Nor are they usually referred yo as "axioms".) Anyway, would you settle for "a defining property" rather than "axiom"? Tito Omburo (talk) 16:45, 22 June 2025 (UTC)

I would suggest "a foundational property": Although the Archimedean axiom is not always used in formal definitions of the real numbers, it is an axiom in all axiomatic definitions of the geometric line (the continuum, in the teminology of the 19th century, the real line in modern language). It is clear that Dedekind had this axiom in mind when defining the real numbers for modeling the continuum. If you do not like "foundational property", one may use also "fundamental property" D.Lazard (talk) 17:04, 22 June 2025 (UTC)

Foundational property is good, but I still think defining property would be more suitable for the basic audience (with axiom even better). (Although I'm also quite happy with Swordfish's latest, with my small copy edit.) Tito Omburo (talk) 17:45, 22 June 2025 (UTC)

My take is that I don't think we need to add flowery language like "fundamental property" or "foundational property", but I'm not going to object to either of those. Mr. Swordfish (talk) 18:27, 22 June 2025 (UTC)

IMO, this is precisely why axiom is better. "Foundational property" feels "flowery", but isn't. Axiom is correct and immediately understandable. Tito Omburo (talk) 18:48, 22 June 2025 (UTC)

The problem with "axiom" is that it assumes a particular axiomatic setup (or at least one of a particular proper subset of possible axiomatic setups) for describing the same thing (the real numbers). The issues being discussed are not specific to the axioms used. Even "foundational property" suggests that you're coming at the question from a foundationalist perspective. I would just go with "property". --Trovatore (talk) 19:14, 22 June 2025 (UTC)

"Axiom" is an ordinary English word. It means, specifically, something whose truth is taken to be self-evident and is not questioned. The word "axiom" does not mean "axiom of set theory". It certainly is an axiom of the real numbers. It does not assume any particular setup. Indeed, as an axiom it is more primitive than even completeness (see Hilbert). If a model of the "real numbers" didnt satisfy Archimedes axiom, it wouldn't be considered the real numbers! Tito Omburo (talk) 21:38, 22 June 2025 (UTC)

It's true that there is a sense of "axiom" that is an ordinary English word, but using that sense in a mathematics article invites confusion, and anyway it's not totally clear that the Archimedean property is an axiom in that sense.

Linguistic aside: Generally after "model of" you name a theory, not a structure. The real numbers are not a theory but a structure. So "model of the real numbers" doesn't exactly make sense in the usual usage, though there are available meanings that I would let you get away with. --Trovatore (talk) 22:21, 22 June 2025 (UTC)

Except that the real numbers existed before anyone formalized them. When they were formalized, if they did not satisfy the Archimedean axiom, that formalization would have been rejected. Thus Archimedes is an axiom. (See the article axiom.) Tito Omburo (talk) 09:35, 23 June 2025 (UTC)

I would say it was a discovered truth about them rather than an axiom. --Trovatore (talk) 17:29, 23 June 2025 (UTC)

It is explicitly on of Hilbert's axioms. Tito Omburo (talk) 17:48, 23 June 2025 (UTC)

Sure. There are probably lots of axiomatizations of the theory of the reals that include it as an axiom. That's a fact about those axiom systems, not a fact about the reals. The property itself is a fact about the reals, but calling it an axiom is not. --Trovatore (talk) 17:50, 23 June 2025 (UTC)

This is an argument that the reals have no axioms then, just properties. Do I understand correctly? Tito Omburo (talk) 17:55, 23 June 2025 (UTC)

Correct. There is no such thing as an "axiom of the reals" separate from a particular axiomatization. --Trovatore (talk) 19:12, 23 June 2025 (UTC)

Whether the real numbers "exist" is a controversial philosophical question. –jacobolus (t) 17:57, 23 June 2025 (UTC)

When I started this discussion I was not aware of any axiomatic presentation of the real numbers that included Archimedes property. This paper provides an axiomatic presentation of the real numbers that does include Archimedes property as an axiom. So, I've learned something here.

But this presentation is a rather unusual approach that leaves out much of the field axioms, replacing them with axioms based on ordering that then imply the usual field axioms. While it's not WP:FRINGE it's not how the axioms defining the reals are conventionally presented.

My take is that referencing such an unusual and unconventional approach in the lead of a general-interest article is not ideal. Whether the real numbers actually exist, or which set of axioms to use to define them is outside the scope of the lead section of this article.

With that said, I'm now of the opinion that referencing the Archimedes property in the lead is an unnecessary complication, and would propose the following third paragraph for the lead:

An elementary proof is given below that involves only elementary arithmetic and the Archimedean property: that there is no positive real number less than the reciprocal of every natural number. There are many other ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. Other proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

— Preceding unsigned comment added by Mr swordfish (talk • contribs) 18:51, 23 June 2025 (UTC)

I would skip the wikilink to mathematical rigor, which seems more distracting than helpful if we are linking mathematical proof immediately afterward. –jacobolus (t) 19:06, 23 June 2025 (UTC)

I have no objections to that. Mr. Swordfish (talk) 19:15, 23 June 2025 (UTC)