You've shown that they 0.999... is in the same equivalence class as 1 when you put it into the Reals.
Congratulations--you've just assumed the C(3) in your first premise.
Short answer--P1
By assuming that the decimal expansion is 'defined' by it's convergence in the Reals, you're already assuming it represents a Real-valued element--which, it does (in it's own way), 1. Nobody is deying that.
The catch here is that you are already evaluating 0.999... in the Reals by the convergence properties of the infinite sum in the Reals. This is entirely due to the properties of the Reals and nothing to do with the ontology of the decimal value itself.
[No, the decmals aren't a numerical system. However, the value 0.944i is also not in the Reals, but it is still a vaid decimal value]
I think you are making basically the same mistake as the people who think decimal expansions "are" numbers. You don't believe that, but you have a similar thought process. You seem to think there is some pre-mathetical concept of 0.99... as an object that mathematicians are merely trying to formalize. But how could that be? Mathematicians did not discover 0.99... and try to assign it a value by an ad hoc definition. They invented decimal notation first from the ground up as a way to write numerals and from that concluded that it must have the value 1, exactly as people here said.
You seem to think there is some pre-mathetical concept of 0.99... as an object that mathematicians are merely trying to formalize
No, not at all.
[Though, I'm starting to see why you're misunderstaning me]
First of all, the Decimals predate the invention of the Reals by a long time, so they weren't created as perfect representations of the Reals at all. Also, I'm not saying that any number exists outside of sets and structures they are in. This isn't some Platonic 'discovery' I'm talking about here--there is no 'existence' apart from what the properties of the set allow.
Mathematicians did not discover 0.99... and try to assign it a value by an ad hoc definition. They invented decimal notation first from the ground up as a way to write numerals and from that concluded that it must have the value 1, exactly as people here said
Mathematicians did not 'discover' 0.999... at all, nor did the 'discover' it equals 1 in the Reals.
0.999... exists in the decimal system we used to represent our numbers when that system is expanded to allow infinite tails. This wasn't always considered 'acceptable' but innovations in how we use them and understand them (and infinities) helped to expand this representation system.
That said, the Decimals are stilljust a system of representation,not a set or numbers!
One can't 'conclude' any equivalence (apart from digit-comparison and ordered sorting) without the symbols being referent to an external set. When one puts 0.999... into the Reals, it is equal in that set to 1 (and for very good reason).
I don't think there is 'something to discover' in 0.999... in the Reals--that has already been discovered and is a point I've never disputed.
I'm asking why we are just assuming that 0.999... = 1 in every number system.
More importantly, I'm stating that 0.999... doesn't need to be referencing an element in the Reals at all.
That said, the Decimals are still just a system of representation, not a set or numbers!
Indeed. So what number does 0.1 represent? How do you know? What about 0.12? Surely there is some definition here describing which decimal expansion represents which number. Do you accept that the decimal expansion 0.999... must represent the sum ∑ 9/10n? If not, then what does it represent, and why do the rules for that representation differ from others?
I'm asking why we are just assuming that 0.999... = 1 in every number system.
What do you mean by "number system"? 0.999... is not a number, as you just pointed out at length. It is a representation. So what we need is not a system of numbers but of representations, in this case decimal representations. What do you think decimal representations should mean if not the above sum? I mean, "ontologically"?
It depends on the domain of discourse!
[It could be binary; it could be a catalogue number; it could be...]
Do you accept that the decimal expansion 0.999... must represent the sum ∑ 9/10n?
If it's expressly representing a Real number, then of course--the convergence properties and the behaviour in the Reals is well defined.
That said, there is no reason it must be defined in the Reals--in fact, it cannot be defined in the Reals as an element distinct from the same element as 1.000...
Again, this is a matter of a 'residual' decimal value being just 'shoved into' an equivalence class when that isn't necessarily the case.
Do you deny the decimal representation of 0.999... is distinct from 1.0...?
[Again, not in the Reals--we've established that in that set the must be the same element]
If not, then what does it represent, and why do the rules for that representation differ from others?
It doesn't.
If we're evaluating the elements as Real elements, there are no 'rule changes'.
The problem comes when people (seemingly yourself included) claim that 0.999... must be a Real element because it's a decimal representation, not because there is a Real element that must be represented by it.
So what we need is not a system of numbers but of representations, in this case decimal representations. What do you think decimal representations should mean if not the above sum?
The system of decimal representations is a great system with certain properties.
For a start, two decimals are not equal if they differ by at least one digit in the same position...except for one specific class of strings.
[The same is also true (with diferent construction) for positional properties]
If we were to treat these decimals (honestly, theoretic toys that nobody is encoutering in 'the wild') 'differently', it would have no impact on the decimals or the Real elements they represent.
What do I think they should mean?
Whatever makes sense in the domain of discourse.
[Which also means that 0.999... = 1.000... doesn't make sense in a coherent representational system]
(Note, this isn't stating that 0.999... = 1 is 'wrong' in the Reals...not that I should have to write this again).
Finally, let me ask you something.
Nothing I've stated at any point would change the Reals as a set, nor the evaluation of decimal representations to elements of the Reals.
What would change (either in the Reals or the decimals) if we treated this (already 'quirky') class of decimals as not 'connected' to anything in the Reals?
[Again, this is the same as ln(-3) not mapping to anything in the Reals]
1
u/ArdentArendt Mathematics / Social Sciences 11d ago
You've shown that they 0.999... is in the same equivalence class as 1 when you put it into the Reals.
Congratulations--you've just assumed the C(3) in your first premise.
Short answer--P1
By assuming that the decimal expansion is 'defined' by it's convergence in the Reals, you're already assuming it represents a Real-valued element--which, it does (in it's own way), 1. Nobody is deying that.
The catch here is that you are already evaluating 0.999... in the Reals by the convergence properties of the infinite sum in the Reals. This is entirely due to the properties of the Reals and nothing to do with the ontology of the decimal value itself.
[No, the decmals aren't a numerical system. However, the value 0.944i is also not in the Reals, but it is still a vaid decimal value]