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In topology the rationals are not discrete[1]: every open interval around zero contains other rational numbers! Also, what do you mean by "rationally-approxima
by red_trumpet 4y ago
In topology the rationals are not discrete[1]: every open interval around zero contains other rational numbers!
Also, what do you mean by "rationally-approximable"? I think the construction of the reals using Cauchy sequences is /exactly/ meant to formalise this.
[1] https://en.wikipedia.org/wiki/Discrete_space https://en.wikipedia.org/wiki/Discrete_space
- andrewla 4y agoThe problem is that Cauchy sequences are too powerful. The intuitionist/constructive version of "rationally approximable" requires a procedure to approximate a number to any requested degree of precision, but cannot guarantee that equality is decideable. My use of discrete was informal and in response to the thread here which equated discarding the notion on the continuum with saying the universe is "discrete". More precise in the topology sense is that the rationals are not compact