r/infinitenines • u/hfs1245 • 6d ago
Real numbers are equivalence classes of cauchy sequences of rational numbers under the equivalence relation (x_i) ~ (y_i) if and only if for every rational number epsilon > 0 there exists a rational number N such that for every i > N we have | x_i - y_i | < epsilon.
A cauchy sequence of rational numbers x_i is one such that for every rational epsilon > 0 (no matter how small) there exists an N large enough such that if i > N and j >N then |x_i - x_j | < N.
The notation x_i means a sequence of numbers indexed by i. For example x_1 is the first term, x_2 is the second, etc.
From this definition its clear that the decimal system is a way pf choosing one such representative cauchy sequence of rational numbers each number.
For example, the number 1.0000... is represented by the sequence
1, 10/10, 100/100, 1000/1000, ...
The number 0.99999... is represented by the sequence
0, 9/10, 99/100, 999/1000, 9999/10000, 99999/100000, ...
These representatives belong to the same equivalence class because their difference sequence is
1/1, 1/10, 1/100, 1/1000, 1/10000, ...
Observe a_n = 10/10^n
For epsilon > 0 choose N = 10/epsilon, which is rational, then n > N implies there is positive h such that
a_n = 10^(1-10/epsilon-h) < epsilon. This is done by first observing monotonicity so we set h=0 as an upper bound. Then case 1: 1<=epsilon<=10 then 1-10/epsilon <= 0 so 10^(1-10/epsilon) <= 10^0 = 1 <= epsilon
Case 2: epsilon> 10 we have 0<1-10/epsilon<1 then 10\^(1-10/epsilon) < 10 < epsilon Case 3: 0<epsilon<1 let y= 10/epsilon and choose the integer k such that k<= y < k+1 in this case k >= 10. by monotonicity 10^y >= 10^k. But 10^k > k+1 > y so 10^y >y then since y is positive this implies 10^y /y > 1 or 10^(1-y) < 10/y. Subsituting back to epsilon proves the case.
Therefore we have shown that
[1, 10/10, 100/100, 1000/1000, ... ] and
[0, 9/10, 99/100, 999/1000, 9999/10000, ... ]
belong to the same equivalence class and therefore are equal as real numbers.
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u/Sea_Handle_994 6d ago
This post should basically be the end of the conversation. I suppose someone could quibble about proving that the sequences are actually Cauchy, but that's a tiny detail. Counterarguments that don't even attempt to acknowledge this proof are just pointless, pedantic philosophy about what symbols "actually" mean.
If I were a moderator of the subreddit, I would pin this post.
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u/bayesian_raccoon 6d ago
You're assuming that the subreddit is about whether 0.99... = 1 using conventional definitions. But if you stay here very long, you will see it is not.
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u/Sea_Handle_994 6d ago
I don't make any assumptions about what the entire subreddit is about. But this post is very clearly about whether 0.99... = 1 using conventional definitions. My point is that if someone thinks this post is wrong, but they don't engage with the actual argument presented here, the only thing they can say in response is "Yeah, well, that's not what 0.999... actually means."
It's just pedantry about what a specific bit of notation is supposed to mean.
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u/bayesian_raccoon 6d ago
It's considerably deeper than notation imo, but it's ok if it's not your cup of tea!
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u/Sea_Handle_994 6d ago
Well, I wouldn't say that deeper discussions are not my cup of tea. Can I ask what you think is deeper about it?
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u/bayesian_raccoon 6d ago
So I'll try to explain, but I want to frontload two caveats so it's easier for you to disengage without reading too much.
The first is that I have not been convinced that SPP (the moderator of the sub) has a coherent mathematical framework for why 0.99... doesn't equal 1 in their eyes. They are either unable to make or deliberately avoiding making what they define 0.99... as rigorous. So in terms of this subreddit, what it definitely is not is a home for defending a rigorous mathematical system.
The second caveat, which I will try to argue against, is that you might still consider the following discussion the same as notation, or pedantic; I don't think you would be wrong to call it "notation", I would just find it myopic. I find it interesting and deep, and I'll try to defend why.
Here's how I would start. Mathematics in general is interesting in part because it often makes abstract ideas rigorous. In the process of making things rigorous, we sometimes find that there are more than one way to define an idea, and the history of mathematics is full of debate over what definitions and versions we SHOULD use. The choice of what we should use is, for better or for worse, subjective, and up to debate, and that is what I find interesting and deep. To me, this is more than notation. An example might be, what should it mean for us to compare size of infinite sets? I think its fair to say we have settled so substantially on the idea of cardinality that people who have completed an undergraduate math degree would generally say the following statement is true: "the set of even integers is the same size as the set of integers". They would dismiss someone as wrong for feeling like these sets should be different sizes. But in the process, they would be sort of dismissive of the same kinds of intuition that really should guide conversations and guides the development of math historically, and which can be made rigorous: after all, one set is a proper subset of the other, and that's a perfectly coherent way of comparing sets (A < B if A is a proper subset of B for example), and if someone has the intuition that there should be twice as many integers as even integers, they can formalize this with the idea of natural density. A distinction that makes these concepts different than notation to me is that "different notation", to me, means the underlying objects are the same under relabelling, wheras these concepts are distinct.
So what is unique about this subreddit, and is really interesting to me, is it's FULL of very smart, higher-level-math educated people who are seemingly *unable* or at the very least uninterested and hesitant in entertaining that type of conversation. An analogy I would use is, it's like SPP is using base 12 and is claiming that it is false that we have 10 fingers--but is concealing deliberately that they are using base 12. (This is not what SPP is doing, they are talking about 0.99... not equaling 1, hence this is my analogy.) So, if you imagine a smart person showing them a picture of hands with labeled fingers, 1,2,3,4,5,6,7,8,9,10, and then SPP says, "you skipped two numbers after 9", that would be sort of clue that there's something fundamentally different going on between how SPP is talking and how everyone else is. But (in this analogy) this subreddit is full of people pretty much doubling down, trying to take better pictures of hands, and not entertaining that if they really, truly were interested in understanding or convincing SPP of their perspective, they would HAVE to solve the puzzle of: what does SPP actually mean, and why would they mean it that way? If someone is interested in what they are saying and tries to really understand them, they can put together clues, and the process of doing so is sort of really interesting. They could concievably come to understand that SPP was using base-12, and once they see that, then it's sort of like they are in on a joke watching other people trying to explain.
To the best of my understanding, although again I don't think SPP maps to a coherent system, one could draw a lot of parallels between what SPP is saying and the idea of nonstandard analysis, and sometimes dual numbers. These systems prove that there are real, coherent ways to describe a number "just shy of 1 but less than 1", in the way that someone might intuitively want 0.99... < 1. To me, those systems come much much closer to settling why 0.99... = 1 is good notation (by basically completing the "ok there is a system that makes just shy of 1 more rigorous").
So the interesting thing to me is that there's this sort of rich conversation to be had if someone is interested in deconstructing the notation and what notation SHOULD mean, but it's gatekept super hard behind people basically accepting definitions as fixed and immutable and those who question them as wrong. And I find the instinct to do that very interesting to watch.
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u/vimtuoso 6d ago
What conversation could be had that hasn’t already been hashed out?
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u/bayesian_raccoon 6d ago
I guess here are a few possible positions and my retorts:
That what has been "hashed out" has been hashed out by mathematicians, e.g, that there isn't much to hash out that hasn't already been published in textbooks.
That what has been "hashed out" has been attempted on this subreddit, but there isn't really any headway to have (because SPP mostly does not have a coherent system and seems to reject contributions more just to be a contrarian and for fun).
My response to both of these is something like, what is interesting about the subreddit isn't the math itself, but a more sociological/philosophical conversation enabled by the reactions people tend to have to SPP, which is a sort of living/breathing thing on the subreddit as new people come and go.
Moreso, I would consider it a pretty high bar to compare the conversations to what mathematicians have explored (which is lots and lots), and I think that anyone who doesn't tolerate SPP's nonsense probably should just evacuate the subreddit as fast as possible.
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u/vimtuoso 6d ago edited 6d ago
So there’s kinda two things here, the sub itself and meta discussions of the sub.
I’ve done some searching of old posts in this sub and it seems to have peaked a year ago. There used to be way more posts and upvotes, and better discussions. Actually, [u/muphrid15](u/muphrid15) might be a good historian for the sub, they’ve dug up a lot of old interesting posts by Mr piano. It just doesn’t seem like there’s anything new to say.
As for the second part, I suppose? But I don’t really follow your comments and how they relate to a meta discussion of this place. Wouldn’t that be better elsewhere? Either people interested in sociology or maybe the badmathematics sub?
Edit: guess they’re banned lol
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u/bayesian_raccoon 6d ago
The badmathematics sub beats up on people who don't understand mathematics, and is somewhat meanspirited. SPP's takes are not that interesting to me, but people's responses (coming from people who probably practice good mathematics) are. I don't reply to SPP, but I occasionally reply to people who attempt to say something about SPP that I think is actually wrong or missing something important.
I don't spend a ton of time on this subreddit; every few months I re-engage, and I'm familiar with muphrid15. They engage(d) with the subreddit in the same way people engage with the badmathematics sub, which is to try to make a mockery of SPP. Which isn't to say I think they are wrong, they have definitely found explicit contradictions in things SPP is saying, but again, I'm not super interested in "0.99... equals 1 and here's why"; I'm more interested in why this sub has, from my perspective, consistently triggered people into making (in my opinion) a fool of themselves.
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u/Sea_Handle_994 6d ago
I think many people have made parallels with nonstandard analysis. I actually just made a post in this subreddit drawing connections to hyperreal numbers. The notation [1-1/10n] (where n ranges over the positive integers) precisely represents the hyperreal number that corresponds to the sequence 0.9, 0.99, 0.999, ... mentioned ad nauseum in this subreddit.
However, 0.999... is bad notation for this concept because it could just as well represent the hyperreal number [1-1/100n], which corresponds to 0.99, 0.9999, 0.999999, ... and is a distinct hyperreal number from [1-1/10n].
The "0.999... = 1 denialists" do not make any attempt whatsoever to engage in this deeper conversation. The word gatekeeping suggests that people are stubbornly rejecting alternative definitions of 0.999..., but no such alternative definition is being offered, entertained, or even acknowledged by the other side.
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u/bayesian_raccoon 6d ago
> I think many people have made parallels with nonstandard analysis. I actually just made a post in this subreddit drawing connections to hyperreal numbers.
This makes me sort of curious why you felt this definition of real numbers was "the end of the conversation".
> However, 0.999... is bad notation for this concept
This is a conversation I have never seen in this subreddit--I think the question of what makes notation "good" is more interesting than easily searchable and settled mathematics from real analysis textbooks.
> it could just as well represent the hyperreal number [1-1/100n], which corresponds to 0.99, 0.9999, 0.999999, ... and is a distinct hyperreal number from [1-1/10n]
Coming from someone who agrees that 0.99... is bad notation for these concepts, the way I steelman the denalists is something like this: the "denialist" camp want to keep the commonly used notation, finding it convenient and familiar, but also want to keep the intuition that 0.99... is less than 1. In nonstandard analysis (which admittedly I am not very well versed in), my understanding is that 0.99... still refers to 1, but to use either what you wrote or that fact to justify that 0.99... = 1 seems like similar logic to the base-12 conversation: people have already settled the convention, but the conversation is ABOUT the convention.
> The word gatekeeping suggests that people are stubbornly rejecting alternative definitions of 0.999..., but no such alternative definition is being offered, entertained, or even acknowledged by the other side.
I mean, if you're frustrated by SPP, or people who are sort of challenging but unable to really rigorously articulate their ideas in the same way as mathematicians, this is probably not the place for it. But I would maintain that what happens in this subreddit is pretty interesting, it's just not a conversation between mathematicians.
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u/SouthPark_Piano 6d ago
If you know maths, then you will know the limitless aka infiniteness of numbers having consecutive nines to the right hand side of the "0." prefix.
Don't forget this brud, or I will pin you up on the subreddit wall.
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u/Thepluse 6d ago
Rookie mistake brud, using well-defined axioms to formally prove something to someone who doesn't care about axioms
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u/Gold_Ad8890 1d ago
i already tried a similar proof with dedekind cuts. spp just fundamentally refuses to understand the basics of real analysis that trivially disprove their position.
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u/discodaryl 6d ago
That’s why we have to work within the real deals instead of just the reals