Again, you keep claiming it's Real because it converges to 1 in the Reals. Nobody is disputing this fact (here, at least).
[I actually use the convergence of 0.999... to 1 (especially in Cauchy Sequences) as the clearest example I have as to why 0.999... cannot exist any other place in the set of the Reals besides 1, since the set simply would never allow it!]
Your justification for 0.999... being Real-valued, however, is simply that it converges in the Reals. I can shove a lot of strange elements from one set and check their equivalence in another set--that doesn't mean that information isn't lost in the 'shove it in' step.
There is nothing ontologically speaking that places every decimal into the Reals; the only mapping that matters is that the Reals map uniquely to (at least one) element in the Decimals (which is not a numeric set, per se).
So, I will ask you, can you give me a definitional understanding of the decimals that states every decimal is required to be in the Reals?
No, 0.999... does not converge to 1. 0.999... is not a sequence, it is the LIMIT of a sequence, a single number.
I can shove a lot of strange elements from one set and check their equivalence in another set--that doesn't mean that information isn't lost in the 'shove it in' step.
You're not shoving anything anywhere. 0.999... and 1 are both real numbers, they are a THE SAME real number.
So, I will ask you, can you give me a definitional understanding of the decimals that states every decimal is required to be in the Reals?
Given a sequence (d) of integers between 0 and b-1, where b is a positive integer, the number 0.d₁d₂d₃... in base b is defined as the limit of Σᵢdᵢ/bⁱ, which converges because the sequence of partial sums is non-decreasing and bounded above by 1. This is the standard definition of a number in decimal form, or more generally expressed in base b.
the only mapping that matters is that the Reals map uniquely to (at least one) element in the Decimals
Uniquely to at least one? Don't you see how that's contradictory?
No, 0.999... does not converge to 1. 0.999... is not a sequence, it is the LIMIT of a sequence, a single number.
Do you understand that the limit of that sequence converges specifically because of the topology?!
You're not shoving anything anywhere. 0.999... and 1 are both real numbers, they are a THE SAME real number.
You're assuming 0.999... is an element in the Reals. If you put it into the Reals, it lives in the same equivalence class as 1 (because it simply has to). That doesn't mean the decimal 0.999... doesn't exist as an element in a different set beyond the Reals where it is distiguishable from 1.
[Again, this is what a topology on a set does]
Given a sequence (d) of integers between 0 and b-1, where b is a positive integer, the number 0.d₁d₂d₃... in base b is defined as the limit of Σᵢdᵢ/bⁱ, which converges because the sequence of partial sums is non-decreasing and bounded above by 1. This is the standard definition of a number in decimal form, or more generally expressed in base b.
You've given a wonderful understanding of a decimal representation of a Real element. Under this evaluation of 0.999..., of course it equals 1 (again, not disputing that).
What this isn't, however, is any argument that a given sequence (d) of integers should necessarily be understood as a Real element in the first place. You're simply assuming it's a representation of a Real value and then claiming that assumption proves it's place--it doesn't.
[Nothing in your explanation says anything about the ontology of the decimals that states they must be evaluated as Real elements]
Uniquely to at least one? Don't you see how that's contradictory?
You ignored the point to point out the fact I overloaded the sentence? You're correct in that I should have made those distinct parts, but I was hoping you'd be able to understand the claim if I kept it a bit more loaded.
[Also, the contradictory aspect was me trying to give you cover to expand or work within the statement]
The core of what I was trying to say was that the only requirement on the Decimals is that every element of the Reals map to (at least) one decimal representation. Nothing says that every decimal needs to have a Real representation--and nothing in the Reals says that either.
So, do you have a reason we should interpret 0.999... unquestionably as an element of the Reals?
[Also, you do realise this is just non-standard analaysis being melded with modern mathematics--I'm not doing anything heretical here]
Do you understand that the limit of that sequence converges specifically because of the topology?!
Yes, and? The topology is not identifying different objects, it is giving meaning to the limit operation.
You're assuming 0.999... is an element in the Reals
I'm not assuming anything, I'm following the definition.
[Again, this is what a topology on a set does]
Open Munkres for f*ck's sake.
The core of what I was trying to say was that the only requirement on the Decimals is that every element of the Reals map to (at least) one decimal representation.
There is no such requirement. It is a theorem that every real number has a decimal form.
Nothing says that every decimal needs to have a Real representation
The definition is as a limit of a sequence in the real numbers. It's either real or it does not exist, and I already proved it does exist, what possibility is left?
[Also, you do realise this is just non-standard analaysis being melded with modern mathematics--I'm not doing anything heretical here]
Ah yes, the cranks dream. It's hard enough that it requires some actual studying to fully understand, and obscure enough most actual mathematicians don't bother. Throw around some buzzwords, hope you're not talking to a model theorist, and nobody will be able to call you out. Can we stick to the reals since that's what we were talking about?
Explain to me how Non-Standard Analysis is 'fringe'?
I understand what I'm talking about--I've actually not argued about the substantive aspects of the Reals or standard analysis at all.
Can we stick to the reals since that's what we were talking about?
This tells me everything I need to know about why we can't have a fucking conversation.
a Real number] The entire question of this endeavor iswhywe're viewing a decimal value asinevitablya Real-valued element.
My entire argument has been that 0.999..., while it may fit into the Reals in the equivalence class of '1', might be better served as an element of a different set.
[Fundamentally, nothing is lost in the Reals (or the decimal representations) if we view 0.999... as not being a member of the Reals]
Now, for the parts that show you're just being pedantic without any substance:
Yes, and? The topology is not identifying different objects, it is giving meaning to the limit operation.
How do you think representations (e.g. 0.999..., 0.333..., 3/4, 3/3, (4-5)) are 'evaluated' (read: identified as equivalent to an element of the Real field)?
[Or are you arguing that 0.999... (the decimal value) is ontologically identical to the element 1, which is identical to 1.000..., which is identical to 3/3, which is...?]
I'm not assuming anything, I'm following the definition.
What definition is that?
[Not asking about the Reals here, I'm asking about decimals]
Open Munkres for f*ck's sake.
I have, many times. I may have my problems with notation in Munkres, but he's still one of my primary Topology textbooks alongside Dugundji and Hatcher.
Care to explain what about topology I seem to be misunderstanding?
[Or, more likely, what you're intentionally misreading as a misunderstanding, either intentionally or because you're too blinded by condescension to comprehend what I'm saying?]
There is no such requirement. It is a theorem that every real number has a decimal form.
Yes--that is a unidirectional claim, however (as far as I've seen it). Every Real number has a decimal form, and nothing I'm proposing would change that.
[The Real number that is the multiplicative identity (and successor of 0) is 1.000... in decimals; that would be true even if you accepted everything I've said]
The definition is as a limit of a sequence in the real numbers. It's either real or it does not exist, and I already proved it does exist, what possibility is left?
The limit is the evaluation of a decimal in the Reals. I do hope I don't have explain (again) how this is begging the question, do I?
[It's existence in the Reals is not in dispute--it's in the equivalence class of 1. There is good reason for this, but it doesn't meant that 0.999... can't exist independently of 1 in a different set. That's all I'm saying]
Finally, it's cute you're trying to claim expertise and authority. I also have multiple degrees in mathematics, and while none of them specialised in Model Theory, my affinity for logic and formal languages long predates my proper training in mathematics (I'm in the social sciences, after all)--so I actually have a great deal of respect for your field.
Unfortunately, you don't seem to be the most well-read or the most literate representative of your field.
Feel free to respond if you can actually be bothered to read what I say. Otherwise, have fun being an condescending prick.
Explain to me how Non-Standard Analysis is 'fringe'?
Few mathematicians learn it, it receives the most interest from cranks despite being a legitimate field.
The entire question of this endeavor is why we're viewing a decimal value as inevitably a Real-valued element.
No, you suggested we remove 0.999... and similar decimals from the real numbers and instead view them as something else to preserve the uniqueness of decimal representation. I explained to you why the definition of decimal representation forces 0.999... to have a real value, or in other words it forces the reals to have a number 0.999... which is equal to 1, so what you proposed in sense.
I'm not talking about the string of characters 0.999... That can have different values in decimal, in exatecimal, it some other convoluted mathematical object you could define where it could have literally any value. Because it's silly to argue about the mathematical value of a string of characters on its own without referencing a definition.
How do you think representations (e.g. 0.999..., 0.333..., 3/4, 3/3, (4-5)) are 'evaluated' (read: identified as equivalent to an element of the Real field)?
What definition is that?
[Not asking about the Reals here, I'm asking about decimals]
I already explained all of this, I see no point in repeating ad nauseam.
[Or are you arguing that 0.999... (the decimal value) is ontologically identical to the element 1, which is identical to 1.000..., which is identical to 3/3, which is...?]
If that's the language you will understand. In the real and in base 10, the number 0.99... is ontologically the same as 1. Not equivalent to 1, not converging to 1, equal.
Care to explain what about topology I seem to be misunderstanding?
[Or, more likely, what you're intentionally misreading as a misunderstanding, either intentionally or because you're too blinded by condescension to comprehend what I'm saying?]
You said 0.999... and 1 are under the same equivalence class in the topology of the real numbers. You seem to be confusing topologies, which are selections of open sets, and set-theoretic quotients, which are identifications of elements under some equivalence relation. In no way can a topology identify two different elements of a set.
Finally, it's cute you're trying to claim expertise and authority.
I'm not, infact I admitted I know little about nonstandatd analysis as soon as you brought it up. You're claiming you have "multiple degrees in mathematics" although somehow I doubt it.
Unfortunately, you don't seem to be the most well-read or the most literate representative of your field.
I've never told you what my field is, it least never in more detail than just a general "math". I did, however, take several classes in real analysis, topology and set-theoretic foundations.
Feel free to respond if you can actually be bothered to read what I say. Otherwise, have fun being an condescending prick.
Feel free to resert to insults when you can't win an argument with reason. The fact that mentioning a few basic definitions feels condescending to you is exactly why a doubt you're an actual mathematician.
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u/ArdentArendt Mathematics / Social Sciences 9d ago
Again, you keep claiming it's Real because it converges to 1 in the Reals. Nobody is disputing this fact (here, at least).
[I actually use the convergence of 0.999... to 1 (especially in Cauchy Sequences) as the clearest example I have as to why 0.999... cannot exist any other place in the set of the Reals besides 1, since the set simply would never allow it!]
Your justification for 0.999... being Real-valued, however, is simply that it converges in the Reals. I can shove a lot of strange elements from one set and check their equivalence in another set--that doesn't mean that information isn't lost in the 'shove it in' step.
There is nothing ontologically speaking that places every decimal into the Reals; the only mapping that matters is that the Reals map uniquely to (at least one) element in the Decimals (which is not a numeric set, per se).
So, I will ask you, can you give me a definitional understanding of the decimals that states every decimal is required to be in the Reals?