The construction of the reals as the Cauchy completion of the rationals was worked out in full and glorious detail in the Montreal spring of 1973. The point is that taking equiconvergence classes of Cauchy sequences fails to construct the reals in a topos because of the absence generally of countable choice. Take your favourite example of a topos in which this fails, and you are most of the way to having your counter-example. To obtain the constructive version of the Cauchy reals, coinciding with the Dedekind reals in any topos with natural number object, you need to think a little more carefully about you are trying to achieve. The important thing about a Dedekind real is that there exist rationals that are arbitrarily close to it. The problem is that of choosing an instance of a rational at distance < 1/n from the real for each n. If you have countable choice, then choose away, get a Cauchy sequence, and have an isomorphism of Cauchy reals with Dedekind reals. Without countable choice, you still have an inhabited subset of the rationals consisting of all rationals at a distance of < 1/n from the Dedekind cut. This gives you a sequence of such subsets - a Cauchy approximation to the real. The constructive version of the Cauchy reals is the set of equiconvergence classes of Cauchy approximations on the rationals. For the details, later extended to the context of seminormed spaces over the rationals, with the set of rationals as the canonical example, see papers such as Burden/Mulvey in SLN 753 and my paper Banach sheaves in JPAA 17, 69-83 (1980). In the present context, the question is whether you wish to study the deficiencies of toposes in which countable choice fails, in which case Cauchy sequences are for you, or whether you want to develop constructive analysis within a topos, in which case you need to look at Cauchy approximations instead. Ask yourself, when you take a point in the closure of a subset, do you get handed a Cauchy sequence converging to it, or a sequence of possible choices of elements within 1/n of it if only you had countable choice to choose them. If the former, go for Cauchy sequences and count your blessings. If the latter, work with Cauchy approximations, which are every bit as powerful as Cauchy sequences and with respect to which the reals are Cauchy complete. Of course, the approach to Banach spaces through completeness defined in terms of Cauchy approximations acquires collateral justification in terms of the approaches to Banach sheaves taken by Auspitz and Banaschewski, to which reference can be found in the papers above. It is also the approach that allows Gelfand duality to be established constructively between commutative C*-algebras and compact completely regular locales in work with Banaschewski and with Vermeulen. Chris Mulvey.
This is a reply to Chris Mulvey's and Mamuka Jibladze's messages. Chris Mulvey's message nicely illustrates Martin Escardo's point that there are different senses in which one might understand Cauchy completion. As Chris confirms, it has long been known that, in toposes not satisfying number-number choice, the Cauchy reals, i.e. the set of Cauchy sequences (with modulus) quotiented by the obvious equivalence, are problematic. Chris takes Cauchy complete to mean complete w.r.t. "Cauchy approximations" which he defines as:
Without countable choice, you still have an inhabited subset of the rationals consisting of all rationals at a distance of < 1/n from the Dedekind cut. This gives you a sequence of such subsets - a Cauchy approximation to the real.
As he remarks, it is well known that the Cauchy reals need not be Cauchy complete w.r.t. Cauchy approximations. Moreover, their "Cauchy completion" is the object of Dedekind reals. Thus, any of the familiar toposes in which Cauchy and Dedekind reals differ (e.g. sheaves on R) provides an example in which the Cauchy reals are not Cauchy complete w.r.t. Cauchy approximations. The above story repeats itself exactly if one changes the meaning of Cauchy completeness to mean completeness w.r.t. a suitable notion of "Cauchy" filter. However, Andrej Bauer was referring to Cauchy completeness in a different sense. A very natural definition of Cauchy completeness is to merely require completeness w.r.t. Cauchy sequences (with modulus) of elements. This is weaker than the definitions above. The open(?) question Andrej referred to is to find an example of a topos (if one exists) in which the Cauchy reals are not themselves Cauchy complete w.r.t. convergent sequences. For this, one of course requires a topos in which the Cauchy and Dedekind reals differ (as the latter are complete). However, the standard examples of such toposes (e.g. sheaves on R) do not answer the question, for, in them, the Cauchy reals do turn out to be complete w.r.t. sequences. One might object to the above question on the grounds that completeness w.r.t. sequences is not the "correct" notion in a topos. There is some validity to this. However, the question originally arose because Martin Escardo and I came up with a definition of an "interval object" (an object of a category corresponding to the closed interval [0,1] in much the same way that a "natural numbers object" corresponds to the natural numbers) that makes sense in the very general setting of an arbitrary category with finite products. When interpreted in Set, the interval object is the interval [0,1]. When interpreted in Top it is the interval with Euclidean topology. When interpreted in an elementary topos, the interval object turns out to be the interval [0,1] constructed within the "Cauchy completion w.r.t convergent sequences of the Cauchy reals within the Dedekind reals", where the quotes are, once again, because the phrase needs careful interpretation. For mathematical details, see our paper in LICS 2001 "A Universal Characterization of the Closed Euclidean Interval". Our approach apparently has something to say related to Mamuka Jiblaze's question. For us the interval is defined as an algebra (implementing an algebraic notion of convexity) freely generated by the object 1+1. In Top, one can replace 1+1 by Sierpinski space as the generating object, in which case the interval with the topology of lower semicontinuity (equivalently the Scott topology) is obtained. Similarly, in a topos, one might take non-decidable objects (e.g. interesting "dominances" in the sense of Rosolini) as generating objects. We have not pursued this latter direction at all, but it might be interesting to do so. Alex Simpson Alex Simpson, LFCS, Division of Informatics, Univ. of Edinburgh Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://www.dcs.ed.ac.uk/home/als Fax: +44 (0)131 667 7209
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Alex Simpson -
C.J.Mulvey＠sussex.ac.uk