Cauchy completions

[C.J.Mulvey@sussex.ac.uk (Christopher Mulvey)]

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.