You are downvoted, but you are actually correct. 0.999... != 1 can be true in nonstandard analysis. So if using standard analysis over nonstandard analysis is a convention, then ultimately 0.999... = 1 is a convention too.
(The Wikipedia article even reproduces that argument)
.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
The Wikipedia article says how, you need a definition of real numbers that includes nonzero infinitesimals (IOW does not satisfy the Archimedian property).
So let there be an ω with 0.999... = 1 - 1/ω. Then a number between 0.999... and 1 would be 1 - 0.5/ω.
How do you know that 1/ω is not 0?
{ 0.999... | 1 } using surreal number notation.
Re:0.999...= 1
by chottocharaii on Tuesday April 28, 2020 @02:33PM
(#23010237)
? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this
Re:0.999...= 1
by MockObject on Tuesday April 28, 2020 @03:11PM
(#23010626)
Either two numbers are equal, or they're not equal. (Unless we're calling into question the law of excluded middle, but must we?)
Re:0.999...= 1
by Dylan16807 on Tuesday April 28, 2020 @03:52PM
(#23011096)
Without using 9999..., please name any two real numbers that don't have a number in between.
Re:0.999...= 1
by momentbruh on Tuesday April 28, 2020 @03:12PM
(#23010639)
That is true for the reals
Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was always < 1.
Fundamentally, 0.999... has no intrinsic meaning, and it’s value depends on the meaning we decide to give this representation.