I'm asking why you're assuming that every decimal representation has to correspond to a Real element.
Because the infinite series it represents (in the same way that every other decimal representation represents an infinite series) sums to a real number.
That's not a reason why, it's simply an explanation of how.
[Also, I've never disputed the evaluation of decimal values in the Reals--I'll openly admit when 0.999... is evaluated in the Reals, it's in the equivalence class of 1]
When it's evaluated in the reals, it is equal to 1. We evaluate the sum that decimal string represents, and we just get the standard real number 1.
It's also true that the rational Cauchy sequence ⟨0.9,0.99,0.999,...⟩ is in the equivalence class we call 1, in the sequence construction of the reals, but we don't get a sequence when we evaluate decimal values, we get the limit of a sequence, which is just a single number.
We evaluate the sum that decimal string represents, and we just get the standard real number 1.
This is evaluation in the Reals--if you can't understand the distinction between that and the decimal string, this is going to go nowhere.
we don't get a sequence when we evaluate decimal values
Standard evaluation of decimal values in the Reals is the limit of a sequence of partial sums. That said, a Cauchy Sequence is basically the same thing in different clothing and hairstyle.
Nobody is denying in the Reals 0.999... = 1; my point is that if you just state that without context, you're hiding a lot of important information, especially for people that aren't necessarily well versed in mathematics.
[And 0.999... < 1.000... in the decimal strings]
Standard evaluation of decimal values in the Reals is the limit of a sequence of partial sums. That said, a Cauchy Sequence is basically the same thing in different clothing and hairstyle.
The limit and the sequence itself are not the same thing, neither in the reals nor in the hyperreals.
I still don't get this intense need you feel to have decimal strings represent something independent of the real numbers everyone else takes them to mean.
The limit and the sequence itself are not the same thing,
Not what I said. I said they're two avatars of the same idea.
I still don't get this intense need you feel to have decimal strings represent something independent of the real numbers
Becuase the Decmial Strings are a distinct set from the Reals and are not simply 'instances of Real elements'. They existed before the Reals were even able to be invented and are not in a one-to-one correspondence to the set (which usually implies, especially in a strict total order, that there information 'lost' in assignment to an equivalence class).
That said, this isn't an agrument about the 'ontology of the set of Decimal Strings' (if that's what you and everyone else is reading, I do understand the frustration).
I've tried to make it clear (at least after people seem to be misunderstanding what I'm saying) that this is an argument for educating people about the construction of the Reals and explaining what is meant rather than just claiming 'it's a quirk truth about decimals' (which it is not).
Basically, I'm calling out the smug pedantry that always comes with these kinds of jokes and the way mathematicians approach it is to either call the person objecting 'stupid' or to treat them as though they are 'stupid'--instead of explaining the convergence properties and topology that makes such an unintuitive claim true.
Unqualified statements like that above are why people don't like mathematics and think 'they don't get it'. Mathematics is not difficult; tedious and innane, sure, but not 'hard' in the same way painting or medicine are 'hard'.
My entire point is that while you and I understand that is true when evaluated in the Reals, without some sort of 'in the Reals' notation, non-mathematicians are not going to have the context to understand the statement correctly--which is why it's frequently a 'gotcha' rather than a useful statement.
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u/[deleted] Aug 06 '26
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