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>nobody can name a system where 1 and 0.999... are unambiguously defined, different numbers. The Hyperreals/Robinson reals. 0.999... < 1 - ε < 1 https://en.w
by ukj 5y ago
>nobody can name a system where 1 and 0.999... are unambiguously defined, different numbers.
The Hyperreals/Robinson reals.
0.999... < 1 - ε < 1
https://en.wikipedia.org/wiki/Hyperreal_number https://en.wikipedia.org/wiki/Hyperreal_number
- cousin_it 5y agoHmm, in the simplest interpretation that seems false, because the hyperreals contain the reals, and 0.999...=1 in the reals. In more sophisticated interpretations, I'm not sure 0.999... unambiguously defines a number. Feel free to correct me, of course.
- ukj 5y agoIt’s definitions/conventions all the way down. Which particular construction/definition of “the” reals/hyper reals are you referring to? The expansion of the “0.n…” notation is a separate matter. It also depends on which particular definition of “=“ and “!=“ you are using. x != y could be the classical “not(x = y)”; or the constructive “x < y OR y < x”. So if 0.999… < 1 is an error, is it a syntax or semantic error? https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...