Definition talk:Real Number

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I've never seen the approach of constructing the real numbers in the interval $[0..1]$.

Why would you do that? The interval isn't closed under $+$. Unless you're suggesting working with $\Z \times [0..1)$? I'm not sure what you mean.

I always knew of the Cauchy, Dedekind and Stevin construction of $\R$ but looking at Wikipedia's sources there are many other approaches as well.

IMHO, I think PW should somehow show a recognition that these approaches exist (to be comprehensive) and then come back to it in the future. Unless of course anyone wants to do it. --Jshflynn (talk) 10:24, 16 January 2013 (UTC)

What page does this comment relate to? --prime mover (talk) 12:00, 16 January 2013 (UTC)
No matter, I understand - this was the result of a faulty template. --prime mover (talk) 12:08, 16 January 2013 (UTC)

Jshflynn, yes, $\Z \times [0,1)$. Thanks to some recent work by I don't remember whom, some of these approaches are now or will shortly be available. --Dfeuer (talk) 15:07, 16 January 2013 (UTC)

Another option is to build the strictly positive reals, the strictly negative reals, and take the disjoint union of those with $0$. That justifies the usual notion of a real number as either $0$ or an optionally signed infinite decimal value. I'll have to dig up the book I have that uses one of these approaches. I think it's a number theory book. --Dfeuer (talk) 16:54, 16 January 2013 (UTC)

Circular problems with the definition using Cauchy Sequences

Many articles about sequences assume we have defined real numbers. Examples: Definition:Convergent Sequence/Note on Domain of N, Definition:Metric Space.

If we don't want to invalidate the construction of $\R$ by Cauchy-sequences, we have to make sure that all definitions and theorems used in its construction do not use $\R$ (better: don't even mention it). This means: when defining an equivalence relation on Cauchy Sequences, this has to be done separately for rational numbers. The theorem for general metric spaces can not be used in the construction of $\R$. --barto (talk) 07:49, 28 January 2017 (EST)

This might be a serious issue. We will have to address this carefully. — Lord_Farin (talk) 04:37, 29 January 2017 (EST)