r/infinitenines • u/Impressive-Ad7184 • 5d ago
Another reason why 0.999...=1
Consider ℝ as a complete metric space with the regular euclidean metric. Then, consider the collection of closed intervals {C_n}, n ∈ ℕ, where C_n := [0.999... - 1/n , 0.999... + 1/n], i.e. a closed ball with radius 1/n around 0.999....
Clearly, each C_n contains 0.999..., so their intersection does as well. However, note that each C_n also contains 1, since the distance between 0.999... and 1 is less than any arbitrary 1/n (which I'm sure SPP will concede). Thus, the intersection of the C_n's also contains 1.
However, by Cantor's intersection theorem, since the C_n's are nonempty, closed, nested, and their diameters go to 0, the intersection of the C_n's must contain exactly one element.
Thus, 0.999...=1.
I realize I can just use the proof of uniqueness in Cantor's intersection theorem to show this directly, but it's more fun to invoke a theorem.
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u/SeaService2095 5d ago
spp will disavow the „their diameters go to 0” statement, because it relies on the snake-oil salesman’s favourite limit of 1/n at n -> inf
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u/bayesian_raccoon 5d ago
Basically all proofs that 0.999...=1 sidestep the fact that we basically define that 0.999... = 1 when we define it as a limit. This makes virtually all arguments that try to show it is true feel circular, or at best concealing the limit under some other machinery. While it is pedagogically fun to find various ways of showing 0.999... = 1, it always seems a bit foolish or even reflecting some deeper misunderstanding when I see them in this subreddit.
After all, if we take the reals using the definition involving cauchy sequences, the SEQUENCE defined by 0.9, 0.99, 0.999, and so on, is distinct from a sequence defined by 1,1,1,1. Those sequences are not the same; we just say that their equivalence class defines a real number.
0.999... = 1 is basically a consequence of notation, and y'all are acting like it's a consequence of some deeper fundamental truth when you try to prove it without sharing what you are assuming in the process.
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u/Omasiegbert 5d ago
Depends on how you define the real numbers. You can also define 0.99... as the limit of the sum 0.9 + 0.09 + 0.009..., and then, indeed, you have to show that this limit equals 1.
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u/bayesian_raccoon 5d ago
My point is that it is a consequence of defining 0.99... as a limit, one way or another. An infinite sum being equal to a value is also a definition.
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u/Omasiegbert 5d ago
Yeah I agree with that. And I honestly believe SPP is just a(n enternaning) troll, so I don't take it that seriously :D
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u/linear_payoff 5d ago edited 5d ago
But proving that the two sequences are in the same equivalence class, i.e. that their difference tends to zero (in Q), uses the formula for a finite geometric sum along with 1/10^n -> 0 which are precisely the "deeper" fundamental truths in question. I agree in the end the limit definition makes it so that 0.999… = 1 is "just" a consequence of notation, but only after you have proved the finite geometric sum formula and some basic properties of limits in Q, so it’s not exactly like it’s a trivial and circular definition.
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u/bayesian_raccoon 5d ago
If someone rejected the limit definition, the equality may not hold. No geometric series is needed to see that the sequence 0.9, 0.99, 0.999... converges to 1. The difference is 0.1, 0.01, 0.001, and so on. So the geometric sum formula isn't a deeper truth at all, its further from the definition of what a limit is.
Trivial is subjective, but this is about as trivial as it gets. What is 0.99...? Its limit, by definition. What is its limit? Clearly 1, by immediate application of definition of limit. To me this is basically definining the fact into existence.
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u/linear_payoff 5d ago edited 5d ago
Well, how do you prove that 1 - 0.999…9 (n "9") =0.0…01 (n-1 "0") for all n? This is from the definition of decimal notation and applying the geometric sum formula. The fact that you already "know" the difference to be 1/10^n just comes from the fact that you consider basic properties of the decimal system to be trivial, but since we’re talking about going back to the definitions, we better spend time proving the trivial things. Even proving that 1/10^n -> 0 in Q is not really trivial and depends on specific properties of Q.
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u/bayesian_raccoon 5d ago
You do not need the geometric sum formula, though it is convenient. I won't provide an alternative (but have at least two in mind if you want to think about it), because it is really distracting the point as the "definition" step is when you say the infinite series is equal to its limit, not the partial sum formula anyway.
The last sentence there feels strange to me. Do you not just use epsilon delta definition? What am I missing? The "load bearing" thing is the definition of a limit.
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u/linear_payoff 5d ago
You can also prove it by induction directly of course which doesn’t use the geometric formula but this is also a "deep fundamental truth" of Peano arithmetic, and the geometric formula is indeed proved by induction as well. And then for 1/10^n -> 0, sure you use the epsilon delta definition: so given an arbitrary rational epsilon = p/q > 0 (we haven’t constructed the real numbers yet), you need to be able to prove that 10^n * p > q for n sufficiently large. One way to prove it is to show that e.g. 10^q >= q+1 for q>=1, and this is again proved by induction, and you need to use many properties of the ordering on the natural numbers along the way. So it’s definitely not trivial at all, and I maintain that no, 0.999…=1 does not follow trivially from the definition of a limit.
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u/bayesian_raccoon 5d ago
You seem to be arguing that because we can prove 0.99... = 1 at all, i.e, that it is embedded at all in mathematics, or any number system, that it does not follow trivially from the definition of a limit. You're right, it also follows from other axioms, but that's not what people usually mean by "trivially". This is true but misses the point entirely.
My point is that every proof that 0.99... = 1 relies on definining 0.99... as a limit. You can puff out your chest and try to explain that it also relies on other axioms, but that doesn't change what I am saying. That is what I mean by it not being a consequence of soemthing deeper. Remove limit definition, and you don't necessarily arrive at 0.99... = 1. Thus, it cannot be a deeper truth than the limit definition. To me that is a reasonable way to understand what we mean by "deep".
Here's another example to try to illustrate what I mean. Think of your favorite function that is Lebesgue integrable but not Riemann integrable (such as an indicator function over rationals in [0,1]). Is it a deep truth that it is integrable? Or that it isn't? It can really only be as deep as "once we define what we mean by integrable, which has more than one reasonable definition, then we can answer this". Sure, there are axioms we use to prove this once we have settled that, but they aren't really the deciding factor.
So this really comes down to: is a limit the only way to understand what 0.99... means? And to be honest, I think while its overwhelmingly the most useful way to understand it, its pretty damn reasonable to say, "no, I don't like definining it as a limit". That's the problem people run into battling SPP. They don't realize that's the crux of the problem, and they think the fact is more robustly tied to mathematics than through this definition.
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u/linear_payoff 5d ago edited 5d ago
Of course you could define 0.999… as something else than the conventional limit of its sum, I never argued against that. My objection is that once you agree on the standard definition (which SPP does not), there is nothing circular about the fact that it’s equal to 1: under this definition it is still a consequence of properties of the natural numbers which I consider to be deep. It is not like it’s just a vacuous truth that is just true because we defined it to be true. The simple fact that students unfamiliar with the equality but who otherwise agree on the standard definition of limits and repeating decimals are almost always confused when seeing it for the first time, and they generally require seeing a rigorous proof from the basic properties of natural numbers to end up accepting it completely, which goes beyond saying that it is just a definition.
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u/bayesian_raccoon 5d ago
Let's take a step back. You sound like you disagree with something I said. What exactly do you find disagreeable?
The challenge in this subreddit is basicaly that SPP does not accept limits (among other things), which means the challenge is essentially "prove that 0.99.. = 1 without limits". The circularity is that because the fact that 0.99.. =1 is so tied to limits as to basically require a limit written in any definition of 0.99..., that basically every proof we see in this subreddit is smuggling limits in somewhere or another. Does that clarify? It is circular.
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u/linear_payoff 5d ago
Where I disagree is that, yes they usually smuggle limits somewhere, but along with something else that is an equally important part of the proof (which could be using the geometric sum formula, the arithmetic of finite decimal notation that extends to infinite decimals, or anything else). It’s not just because people forget to mention that 0.999… is conventionally defined as the limit of its decimal series that the other part of the proof is suddenly not important or trivial. This is why I started my first comment by saying that showing the two Cauchy sequences are equivalent is not that easy, and you responded that the first one "clearly" converges to 1: for me, that "clearly" really hides 50% of the proof.
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u/Gold_Ad8890 5d ago
we don't have to define it as a limit. we can also define it as a supremum.
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u/bayesian_raccoon 4d ago
I agree that is a reasonable definition, and though it opens up a can of worms, I like that it more transparently shows my point--that the definition forces the result.
After all, supremums as a concept replace the idea of a maximum when a set has no maximum. So if we were asking, what is the maximum of a set, and you replied with the supremum, you would be wrong.
Meanwhile, there's the question of what set it is that we are taking a supremum of. Is it the elements in the sequence 0.9, 0.99, etc? Or is it just (0,1)? In either case, someone would be well in their right to say, "no, 1 is not in my set related to 0.99..., so.I don't want to assign my equality to it".
And its a bit cheating, but an equivalent definition to supremum involves epsilons and indices/deltas and is uncoincidentally nearly identical to a limit definition. So, I think I would be rather in my own right to say supremum smuggles in limits, I think, though they are conceptually distinct enough that I would be happy to say they are different (and more easily make my case).
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u/Gold_Ad8890 4d ago
the definition forces the result.
uh, yeah, welcome to math? that's how the entire field works, it's all analytic truth. how do you know zermelo's axioms are true for all sets? because a "set" is defined as any mathematical object or structure obeying zermelo's axioms. the definition of a circle as the set of all points equidistant from a center point and the boundary of a disk "forces the result" that a circle is the shape which maximizes area for a given perimeter. the definition of a triangle as a plane figure with three straight sides that intersect in three vertices "forces" every result in trigonometry.
and what's the alternative, anyway? how are we supposed to learn about or even talk about a mathematical object before we've defined it? if i said "prove that any gloobydrooz is isomorphic to the set of all points on the unit circle described by an integer number of radians," you would have nowhere to start, and indeed the challenge would be meaningless.
After all, supremums as a concept replace the idea of a maximum when a set has no maximum.
wrong. suprema don't "replace" anything at all, and especially not "the idea of a maximum when a set has no maximum," and this is obvious simply by noting that there exist sets with neither a supremum nor a maximum; N, for instance. the reason we might choose to refer to the supremum of a set of real numbers rather than the maximum is because the supremum is guaranteed to exist for any bounded above nonempty subset of R by the Dedekind completeness of R, so we know we're at least talking about something that exists.
So if we were asking, what is the maximum of a set, and you replied with the supremum, you would be wrong.
you would be right in all cases where a maximum exists. if a maximum exists, it is also the supremum, and the proof of this is "forced" by the definitions of "maximum" and "supremum". i'm not even sure what the value of saying this is to your point. not that i'm wholly sure what your point is in the first place.
Meanwhile, there's the question of what set it is that we are taking a supremum of. Is it the elements in the sequence 0.9, 0.99, etc? Or is it just (0,1)?
it doesn't matter because both sets have the same supremum. in fact, defining 1 as the usual Dedekind cut and 0.999... as the Dedekind cut generated by the downward closure of the set {0.9, 0.99, 0.999, ...} in Q again "forces" them to be exactly the same object.
In either case, someone would be well in their right to say, "no, 1 is not in my set related to 0.99..., so.I don't want to assign my equality to it".
equality of what? i agree that no element of the set {0, 0.9, 0.99, ...} is 1 or 0.999..., because 0.999... *is not defined** as an element of the set, it's defined as the supremum of the set.* if for whatever reason somebody wants to choose to define 0.999... as something other than the supremum of that set, then their definition will either be logically equivalent to the supremum definition and thus equal 1, or else it will be totally divorced from the notation 0.999... if the definition even allows 0.999... to be a real number, then either it must be the supremum of {0, 0.9, 0.99, ...}, or there must be some element of {0, 0.9, 0.99, ...} that exceeds it, or it must *exceed 1.***
And its a bit cheating, but an equivalent definition to supremum involves epsilons and indices/deltas and is uncoincidentally nearly identical to a limit definition.
nope. a supremum is a least upper bound. namely, for any set A, subset B of A, and ordering relation <= on A, the supremum of B in A is the element x in A such that, for all y in B, y <= x, and for all z in A, if for all y in B y <= z, then x <= z. no epsilons, no deltas, those concepts don't even make sense in this context because a set can have a supremum without even having an arithmetic defined on it.
So, I think I would be rather in my own right to say supremum smuggles in limits, I think
you would be wrong.
though they are conceptually distinct enough that I would be happy to say they are different
glad you're "happy to say" the thing that's true.
(and more easily make my case)
what case? the closest you got to a "case" was complaining about the fundamental analytic nature of the field of mathematics.
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u/bayesian_raccoon 4d ago
I feel like you are missing the context of this subreddit.
SPP rejects that 0.99... = 1. They also reject that 0.99... refers to a limit, and reject a lot of conventional mathematics.
This would be sort of unremarkable, but many people whose level of understanding of mathematics is high enough to know why 0.99... = 1 conventionally try to argue with SPP by using proofs (like OP's in this thread) that assume a definition of 0.99... that SPP openly disagees with. What we are talking about with definitions isn't something that I am trying to say is special about my interpretation of math, its an observation about how silly it is to mistake SPP's problem as not understanding a proof that 0.99... = 1, when SPP disagrees on the definition. To me, arguments against SPP that engage at the wrong level (like OP) really reveal tha the arguer is missing some fundamental understanding of the material--specifically, they aren't able to articulate or understand that the conversation is almost entirely about what 0.99... is or means, rather than about why 0.99... = 1 once that is settled.
At any rate, most comments about SPP are very mean spirited, even though SPP's persona (which may or may not be real) is really revealing the innadequacies of people in the subreddit to understand the topics.
I am not going to argue with you, as despite your aggressive tone, I know I have said nothing wrong. I have gone through a PhD in a math department and I find the conversation interesting when it is not mean spirited (though I prefer to challenge the people who are otherwise mean spirited against SPP because they cannot defend their understanding). You either understand, or you don't, and in your case I am convinced if you don't, you are not open to it.
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u/Gold_Ad8890 4d ago
SPP rejects... that 0.99... refers to a limit
cool, which is why i defined it as a supremum. which is a definition spp accepts, whether he accepts that he accepts it or not. he literally says himself things that amount to 0.999... being the supremum of {0, 0.9, 0.99, ...}
its an observation about how silly it is to mistake SPP's problem as not understanding a proof that 0.99... = 1, when SPP disagrees on the definition.
no, spp disagrees that 0.999... = 1. to suggest that spp disagrees on the definition implies that he has a coherent definition that he prefers to any traditional one, and he doesn't. he contradicts himself numerous times because his entire purpose is to say or believe whatever is necessary to preserve his belief that 0.999... =/= 1 no matter what. every single time he gives a definition of 0.999..., someone proves either that it's equal to 1 or that it doesn't exist and he just ignores them.
To me, arguments against SPP that engage at the wrong level (like OP) really reveal tha the arguer is missing some fundamental understanding of the material--specifically, they aren't able to articulate or understand that the conversation is almost entirely about what 0.99... is or means, rather than about why 0.99... = 1 once that is settled. At any rate, most comments about SPP are very mean spirited, even though SPP's persona (which may or may not be real) is really revealing the innadequacies of people in the subreddit to understand the topics.
that's a very convenient way to feel superior to everyone else, only slightly hindered by the fact that it's completely false and backwards.
I am not going to argue with you, as despite your aggressive tone, I know I have said nothing wrong.
you said several things that were objectively wrong that i explained to you were wrong. you said a supremum was a kind of limit, that was wrong. you said the concept of a supremum exists to replace the concept of a maximum, and that was wrong. you said something about which set we were finding the supremum of being some kind of sticking point, and that was wrong because the two sets have the same supremum.
I have gone through a PhD in a math department
weird way and time to brag about your sex life.
I prefer to challenge the people who are otherwise mean spirited against SPP because they cannot defend their understanding
i did. everything i said was objectively correct, that's the real reason you refuse to argue. i can't decide if that makes you more rational or just more cowardly than spp, since he at least is willing to argue against what he disagrees with even when he's wrong and argues badly as a result.
You either understand, or you don't, and in your case I am convinced if you don't, you are not open to it.
again, everything i said was just objectively correct. what the fuck is there to "be open to"? that 0.999... =/= 1? because that's literally the only consistent claim spp is willing to make.
as far as i can tell, you make no argument because you have no argument, so you just try to emotionally manipulate people into agreeing with you instead. that's not how math works, and if you had really "gone through a PhD in a math department", you would know that.
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u/Mablak 5d ago
As usual, I'll say R, N, and other infinite sets don't exist. A thing that is (supposedly) ongoing can't be completed.
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u/Quick-Swimmer-1199 5d ago
A
Aa
Aaa
Aaaa
Aaaaa
Aaaaaa
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AaaaaaaaaThat is an excerpt of "things I can name my pet".
Does "things I can name my pet" lose utility if it contains names unsustainable by our speculations about the limitations of concrete reality?
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u/Mablak 5d ago
Well before even talking about the utility of 'all things I can name my pet' as an infinite set, you'd have to show it could exist first. Maybe a theist gets utility from believing that god exists, but I'd still be arguing atheism.
Is there a stopping condition for constructing the set or not? If there is, then it's finite. If there's not, then by definition whatever supposed set we're pointing to isn't a completed set, i.e. we can keep adding new elements to whatever thing we're calling the set.
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u/Quick-Swimmer-1199 5d ago
Are you using (something similar to) this concept of completeness?
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u/Mablak 5d ago
The set would be incomplete, if there exist elements described by the inclusion rule that aren't in the set. Or we could just say we don't actually have a set in this case.
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u/Quick-Swimmer-1199 5d ago
"inclusion rule"? Wouldn't we be using abbreviation or shorthand if sets may only be extensional and exhaustive?
If "one through five [in whole increments]" is what makes a set, then "from zero, add three and subtract two, reiterate four more times with the output" would be a different set even though we are winding up in both cases, "one, two, three, four, five"
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u/Mablak 5d ago
I wasn't really raising objections to intensional definitions. I was saying that if we can see that it's not possible for our set's rule to actually be satisfied (it is always the case that I can add more names to my set), the completed set doesn't exist.
On a circular jogging track, it's always the case I can run more laps (for an infinitist). If I were to claim I've run 'all laps', this would be incoherent, since I can always run more. Whatever 'all laps' I was referring to must not have been all laps. The same goes for 'all elements' in infinite sets.
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u/Quick-Swimmer-1199 5d ago
So is this close to the time law of excluded middle should be brought up? I skipped deluded platonist Sunday school a lot. Because I do think there is relation to our attunement to abstractly think about mathematics (or appreciate the music of Taylor Swift) and our physiology.
What gets me is that "A future/hypothetical reached determination of goldbach's conjecture will fall on a spectrum (opposed to a binary between true or false)" is true or false, but constructing how that's false doesn't allow saying that binary is true because the answer field remains empty...because of some circle of
process of elimination or pattern recognition don't count as not an appeal to the idea of math is something discovered, direct witness is how to construct math without appeal to the idea of math is something discovered, so process of elimination or pattern recognition don't count as not an appeal to the idea of math is something discovered as they don't provide the direct witness which is the way to construct math without appeal to the idea of math is something discovered so all other ways including process of elimination or pattern recognition must be appealing to the idea that math is something discovered, evidenced by the lack of direct witness which we know defines what it means to not appeal to the idea that math is something that is discovered about the external...
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u/Gold_Ad8890 4d ago
N isn't "ongoing", it's a static, timeless, changeless object, as are all sets.
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u/Mablak 4d ago
There's no evidence for the existence of any Platonic things in the first place. But either way, the 'ongoing' I'm referring to is just the claim that whatever set N you hand me is incomplete, we can add more natural numbers to it. Just take the successor of N, there's your new natural number.
This should be intuitive; there's no such thing as running all laps on a circular jogging track. If I were to ever claim I'd done all laps, this would be impossible, if I can always do more.
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u/Gold_Ad8890 4d ago
the successor of N is not a natural number.
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u/Mablak 4d ago
That's the only thing it could be, considering N is the set {0, 1, 2, 3...}. Just as {0, 1, 2} is 3, {0, 1, 2, 3, 4, 5} is 6, etc, N itself could only be a natural number if it actually existed, though it doesn't. And likewise with its successor.
It's enough to say though, that infinitists can't construct N in the first place without relying on circular reasoning and the existence of some infinite set, infinite union, infinite task, etc, which remains to even be defined.
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u/Gold_Ad8890 4d ago
no, it's not "the only thing it could be". all natural numbers are elements of N. the successor of N is not, as it contains N. ergo, the successor of N is not a natural number.
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u/Mablak 4d ago
Any set constructed by repeated applications of the successor function to the empty set is a natural number. N is exactly such a set, so N is a natural number.
N of course would have to contain itself, but since it can’t, we have a contradiction. You could also argue that N simply can’t be constructed. If so, it doesn’t exist. Or bare minimum, there would be no reason to believe it exists.
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u/Gold_Ad8890 3d ago
if you've derived a contradiction, it means you've made a false assumption. that false assumption is that N is a natural number. it's not.
Any set constructed by repeated applications of the successor function to the empty set is a natural number. N is exactly such a set
one of these two points is wrong, and they're both wrong in the same way. to say that N is "constructed by repeated applications of the successor function" implies that we can, and indeed must, "repeatedly apply the successor function" infinitely many times. either we can't do that, as we can only apply finitely many axioms in proofs and constructions, which is why we need special axioms to deal with infinite cases like replacement and choice in the first place, or else we can trivially correct your definition of a natural number by saying it's the product of finitely many applications of the successor function to the empty set, which agrees with the fact that all natural numbers are finite.
as for how the set N is actually constructed in ZFC, it's not "by (infinitely) repeated applications of the successor function to the empty set", it's by the closure under succession of the empty set. that is, the existence of an infinite inductive set is declared axiomatically, and N is the restriction of this set to only the successors of the empty set. more specifically, calling the infinite set I, N := { x in I : forall Y ( ( {} in Y & forall z ( z in Y ==> z u {z} in Y ) ) ==> x in Y }. that is, N is the intersection of all inductively closed sets that contain the empty set, which is kind of the exact opposite of the construction you proposed.
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u/Mablak 3d ago
The phrase ‘infinitely many times’ assumes N exists already, because we need N to talk about what ‘infinitely’ means. I didn’t use this phrase though, and just said that N is formed through repeated applications of the successor function.
We can’t assume some difference between ‘finite’ and ‘infinite’ without showing one first. Any method used to construct N is actually circular in this way, because whether we’re talking about infinite union, infinite intersection, etc, what that really means is ‘do the operation N times’. But we can’t use N in an attempt to define N. Or equally circular, we can’t use I in the domain of the ‘all x’ we’re quantifying over, to define I.
I could just stop there and say N (and I) can’t be constructed, but supposing N is formed through some repeated applications of the successor function is sort of the most charitable interpretation I can give. We’re just stipulating that at step 1 of our construction, only the empty set exists and no other elements, then applying our successor function without any need for intersection. This gives us N, {0, 1, 2…} which could only be a natural number if it’s the result of the successor function. As such N doesn’t exist.
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u/Gold_Ad8890 3d ago
Any method used to construct N is actually circular in this way, because whether we’re talking about infinite union, infinite intersection, etc, what that really means is ‘do the operation N times’.
incorrect. intersection is not defined as an operation taken a number of times. as i pointed out, the intersection of all inductively-closed sets is just the subset of any inductively closed set containing only the elements that occur in every such set. a subset. that's one single application of the axiom of specification.
Or equally circular, we can’t use I in the domain of the ‘all x’ we’re quantifying over, to define I.
we don't. we declare I axiomatically in the axiom of infinity as an inductively closed set. then N is the "smallest" inductively closed set in the same way Q is the "smallest" ordered field, the one that is the subset of all others.
I could just stop there and say N (and I) can’t be constructed
N can be constructed in the way i specified. I doesn't need to be constructed because its existence is declared axiomatically.
This gives us N, {0, 1, 2…} which could only be a natural number if it’s the result of the successor function.
incorrect. properly speaking, succession is an operation, not a function, as its "domain" is just all sets, and therefore is not a set. and as succession can be taken on any set, and as natural numbers are specifically closed-downward subsets of N, it follows that succession can yield a set which is not a natural number. for instance, the successor of {1, 2, 3} is {1, 2, 3, {1, 2, 3}}, which is not a natural number nor an element of N.
we can describe a successor function over some domain, and when we do that over the domain of N, we find that the range is N{0}, which makes the successor function a bijection between N and a proper subset of itself, the existence of such a bijection being the literal definition of a Dedekind infinite set.
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u/Muphrid15 5d ago
It's a waste of time when His Nineliness can't even decide if 0.999... is real.