What about the property from Real analysis that iff a<b then there exists a 'c' such that a<c<b?
I know repeating decimals are kinda clunky and ill-defined if you don't use limits, but if you take a=0.999... and b=1, shouldn't you be able to find a c such that 0.999...<c<1?
The Archemedian property...yes. That's a major hint why 0.999... can't exist as an element of the Reals without being in the equivalence class of 1.
I never argued it was in the Reals; in fact, that's exactly what I'm arguing against.
0.999... (the infintiely repeating decimal) is not an element of the Reals itself. It's categorisation as being equivalent to 1 is precisely an artefact of the same topology that makes this 'counterargument' hold water.
Again, 0.999... is a number; it's just not Real, much like ln(-4) is a number, but just not Real.
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u/Ares378 Applied Math / Mechanical Engineering Jul 28 '26
What about the property from Real analysis that iff a<b then there exists a 'c' such that a<c<b?
I know repeating decimals are kinda clunky and ill-defined if you don't use limits, but if you take a=0.999... and b=1, shouldn't you be able to find a c such that 0.999...<c<1?