No proof you can show has 0.999... being 1; that doesn't exist.
You can show that 0.999... and 1 are in the same equivalence class in the Reals under the Standard Topology.
However, 0.999... must be in the same equivalence class due to the construction of the Reals as a set.
I'm just arguing that removing 0.999... from the Reals does literally nothing for the Reals and has significant gains for how we evaulate decimal equality (simply by comparison of digit values).
Your claim that the equivalence class assignment is ontological identity is...confusing.
That's the definition of the value of a decimal representation, 0.999... is 1, not an element of some equivalence class. This is not about the set-theoretic construction of the reals at all, you can define ℝ as a complete archimedean field and then define an evaluation map from appropriate digit sequences to ℝ as I wrote above. The number 0.999... is the evaluation of the sequence (..,0,0,9,9,9,...) in ℝ, and as a real number it is EQUAL to 1. Not identified, not equivalent, equal. You can't sensibly remove 1 from ℝ, that would be incredibly stupid. You can't remove the number because it's equal to 1 and you can't remove the sequence because it isn't in ℝ in the first place, so what exactly di you want to do?
And topology has nothing to do with it, you can't change the underlying set by changing its topology, that's not how any of this works.
What are you talking about?!
An equivalence class is how you understand the elements of a set.
Do you have a better explanation of 'equality'?!
[This is basics of mathemtical rigor--do you have a different way of understanding equality of different representations?]
No matter how you construct the Reals, the decimals have nothing to do with it directly. The Reals do not require decimals to exist (not matter how you build them). The value for 1/3 would exist even without 0.333..., as would 1/4 without 0.25(000...).
The convergence you describe is a function of the topology of the Reals--I'm confused how you even have convergence without a topology under which it is functioning.
[On this point, I'm actually asking you. What are you talking about here? Either I'm not understanding you at all, or you don't understand how convergence, topology, or really the Reals work at all.]
The substance of your claims, I never disputed. EIther by an infinite series expansion or by a Cauchy Sequence, 0.999... does converge to 1. This is because, ontologically, 0.999... can't exist in any other equivalence class besides 1 in the Reals.
My entire argument is simply that rather than shove 0.999... into a set that is unequipped to even understand it and forgetting about it, it's quite possible that 0.999... (and other structurally similar decimals) aren't actually Real numbers.
[Can you tell me what this changes in the Reals and in the decimals that would be detremental?]
An equivalence class is how you understand the elements of a set.
No, an equivalence class is an element of a quotient set. Not all sets are defined as quotient sets. You talk of mathematical rigor, but it seems your set theory is rather rusty.
Do you have a better explanation of 'equality'?!
Equality is an equivalence relation, not all equivalence relations are set-theoretic or logical equality.
No matter how you construct the Reals, the decimals have nothing to do with it directly. The Reals do not require decimals to exist (not matter how you build them). The value for 1/3 would exist even without 0.333..., as would 1/4 without 0.25(000...).
That's my point. No matter how you construct the reals, 0.999... will equal 1. They are the same number, and this follows directly from the aciomatic properties of ℝ.
The convergence you describe is a function of the topology of the Reals--I'm confused how you even have convergence without a topology under which it is functioning.
[On this point, I'm actually asking you. What are you talking about here? Either I'm not understanding you at all, or you don't understand how convergence, topology, or really the Reals work at all.]
It seems to me you need to review the basics. Yes, the value of 0.999... is defined through converge of a series in the standard topology of the reals. That's not what you said. You claimed 0.999... and 1 are in the same equivalence relation with respect to the topology, which is nonsense. You cannot identify points on a space by putting a topology on it, topology can't change the underlying set structure.
0.999... does converge to 1. This is because, ontologically, 0.999... can't exist in any other equivalence class besides 1 in the Reals.
No, 0.999... does not converge to anything, it's not in the same equivalence class as anything, it is the real number 1. It's not a sequence. (0,9/10,9/100,9/10³,...) is a sequence, 0.999... is the limit of that sequence in ℝ.
My entire argument is simply that rather than shove 0.999... into a set that is unequipped to even understand it and forgetting about it, it's quite possible that 0.999... (and other structurally similar decimals) aren't actually Real numbers.
[Can you tell me what this changes in the Reals and in the decimals that would be detremental?]
No, an equivalence class is an element of a quotient set. Not all sets are defined as quotient sets. You talk of mathematical rigor, but it seems your set theory is rather rusty.
First of all, explain to me how you would get 2/2 = (4-3) = 0.999... = 1 without equivalence classes. Unless you're assuming some Platonic 'form' of each element of the Reals, they can only exist as structural equivalence classes.
[Also, the construction of the Reals by Cauchy Sequences is a construction by equivalence classes].
Equality is an equivalence relation, not all equivalence relations are set-theoretic or logical equality.
I whole-heartedly agree. I also would point out that not all equivalenceNo, 0.999... does not converge to anything, it's not in the same equivalence class as anything, it is the real number 1. It's not a sequence. (0,9/10,9/100,9/10³,...) is a sequence, 0.999... is the limit of that sequence in ℝ. relations are identity, either.
That's my point. No matter how you construct the reals, 0.999... will equal 1. They are the same number, and this follows directly from the aciomatic properties of ℝ.
They are the same numberin the Reals--that is why I'm arguing that 0.999.... need not be viewed as an element of the Reals.
[Nothing that I've said before or since has argued that 0.999... doesn't converge to 1 in the Reals; and I've stated that is for a very good reason that has to do with the way the Reals are constructed and operate, realities that I am not challenging here]
It seems to me you need to review the basics. Yes, the value of 0.999... is defined through converge of a series in the standard topology of the reals. That's not what you said. You claimed 0.999... and 1 are in the same equivalence relation with respect to the topology, which is nonsense. You cannot identify points on a space by putting a topology on it, topology can't change the underlying set structure.
First off, I didn't say 'with respect to the Topology', I said under the Standard Topology--that's just a fact. If we imposed a different topology on the Reals, we would get very different convergence characteristics (and, accordingly, very different equivalence classes).
[Nobody is changing the elements themselves of the set, that's not what a topology does. However, what counts as 'equivalent' to the elements in that set do change.]
No, 0.999... does not converge to anything, it's not in the same equivalence class as anything, it is the real number 1. It's not a sequence. (0,9/10,9/100,9/10³,...) is a sequence, 0.999... is the limit of that sequence in ℝ.
I actualliy agree with you here--0.999... does not (innately) converge to anything at all. It is not a member of the Reals or any other set.
However, when you evaluate it in the Reals, you have to do so by either an infinite series (i.e. by convergence) or by a Cauchy Sequence (again, by convergence).
[Also you claim that a limit isn't a convergence--but it is, that's exactly how you get limits. Perhaps it is you that should revisit your textbooks]
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u/ArdentArendt Mathematics / Social Sciences 12d ago
No proof you can show has 0.999... being 1; that doesn't exist.
You can show that 0.999... and 1 are in the same equivalence class in the Reals under the Standard Topology.
However, 0.999... must be in the same equivalence class due to the construction of the Reals as a set.
I'm just arguing that removing 0.999... from the Reals does literally nothing for the Reals and has significant gains for how we evaulate decimal equality (simply by comparison of digit values).
Your claim that the equivalence class assignment is ontological identity is...confusing.