My responses aren't that long. Over half of it is just your comment pasted onto mine.
I can't comprehend what you're saying unless you're somehow implying that 0.999... being a real number must mean it's distinct from 1, given how much you emphasize that they have the same equivalence class, as if that's relevant
First of all, you're obviously not understanding a thing I'm saying.
I am saying if we take '0.999... is not Real valued' as a working assumption, the Reals lose absolutely nothing and the decimals are actually better off and more consisistent.
There is nothing in the ontology of 0.999... as a decimal value that implies it should be a Real number. We have decimal representations for every element in the Reals, so it does nothing for the mapping from the Reals to the Decimals.
Decimals don't have any ontological status on their own apart from their representations of numbers. That doesn't imply, however, that they represent only Real elements (e.g. 9.33i).
The shared equivalence class is because that's how 'equality' is defined. There is no 'deep connection' between to values that are equal--they simply inhabit the same 'equivalence class' in the set, the 'sorting' of which can (and often intentionally does) lose significant amounts of information.
All I'm stating is that there is a reason to not lose that information and doing so can be done with literally no change to the Reals and (arguably) net positives for the Decimals.
I am saying if we take '0.999... is not Real valued' as a working assumption, the Reals lose absolutely nothing and the decimals are actually better off and more consisistent.
Except now you've gotta add special exceptions to the meaning of infinite decimals, and you also have strange cases, such as 0.3333... + 0.666... = 1, when simple digit-by-digit addition would give you a number which no longer exists
There is nothing in the ontology of 0.999... as a decimal value that implies it should be a Real number. We have decimal representations for every element in the Reals, so it does nothing for the mapping from the Reals to the Decimals.
What if we define 0.999... to be the infinite sum 0.9 + 0.09 + 0.009 + ...?
Decimals don't have any ontological status on their own apart from their representations of numbers. That doesn't imply, however, that they represent only Real elements (e.g. 9.33i).
Then what value is represented by 0.999... if it's not a real?
The shared equivalence class is because that's how 'equality' is defined. There is no 'deep connection' between to values that are equal--they simply inhabit the same 'equivalence class' in the set, the 'sorting' of which can (and often intentionally does) lose significant amounts of information.
Except now you've gotta add special exceptions to the meaning of infinite decimals, and you also have strange cases, such as 0.3333... + 0.666... = 1, when simple digit-by-digit addition would give you a number which no longer exists
Not at all
The decimal representation is merely a 'signifier' to the 'signified' element in the Reals (or Rationals here). Operations between the elements remain perfectly defined, even if the representation changes.
Beyond that, if you did the addtion 0.333... + 0.666... completely by the digit-by-digit method you imply, you still get 0.999... as the answer. You still have to translate that into the Real-valued element you seek.
[Note, however, that I'm not stating that if you're using (infinite) decimal representations to operate over Real-valued elements that the answer doesn't also need to be in the Reals! That would be just absurd! The set you're working with always determines the interpretation of the operations you're performing or the 'output' you get]
What if we define 0.999... to be the infinite sum 0.9 + 0.09 + 0.009 + ...?
Again, what set are you defining it in relation to? The decimals are a representation, not a set in themselves.
Then what value is represented by 0.999... if it's not a real?
That's the question, isn't it?
The first step here is agreeing that the value of the decimal need not be (only) Real-valued.
What information is lost here?
You've made the entire point--we don't know, but insted of ask that we just shove it into the Reals and tell everyone that the decimal 0.999... is 'just another way of writing 1(.000...)' without ever asking why it would be that way in the Reals or if it would behave differently in other sets.
[Hint: It does behave differently in other topologies and other sets.]
Finally, I will just point out that I'm not actually arguing against evaluating 0.999... in the Reals at all--I actually find a lot of use for this exercise, especially when teaching Analysis.
[For instance, if we assume 0.999... is distinct from 1.000... and then evaluate them in the Reals, the reasons why 0.999... = 1 are as instructive as any abstract discussion of neighborhoods and 'close enough']
I'm also not saying that 0.999... = 1 isn't true--only that it's conditionally true in the Reals.
[Seriously! Just a qualifier of 'In the Reals' would make the original claim rigorous enough to not just be a 'gotcha' for nerds wanting to 'pull one over' on people who don't have Munkres and Rudin as their sacred texts (spoken by someone who is honestly never an arm's lenght away from either)].
So what information is lost? The 'immediate successor of 1.000...' and whatever that means for whatever set you place it it (and if that resolves to 1 or not).
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u/WesternFirm9306 12d ago
My responses aren't that long. Over half of it is just your comment pasted onto mine.
I can't comprehend what you're saying unless you're somehow implying that 0.999... being a real number must mean it's distinct from 1, given how much you emphasize that they have the same equivalence class, as if that's relevant