Re: Cauchy completeness of Cauchy reals

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Mon, 3 Feb 2003, Vaughan Pratt wrote:
>
> While I'm comfortable with coalgebraic presentations of the continuum, as
> well as such algebraic presentations as the field P/I (P being a ring of
> certain polynomials, I the ideal of P generated by 1-2x) that I mentioned
> a week or so ago, I'm afraid I'm no judge of constructive approaches to
> formulating Dedekind cuts.  Would a toposopher (or a constructivist of
> any other stripe) view the following variants as all more or less equally
> constructive, for example?
>
> 1.  Define a (closed) interval in the reals as a disjoint pair (L,R)
> consisting of an order ideal L and an order filter R, both in the rationals
> standardly ordered, both lacking endpoints.  Order intervals by pairwise
> inclusion: (L,R) <= (L',R') when L is a subset of L' and R is a subset of R'.
> Define the reals to be the maximal elements in this order.  Define an
> irrational to be a real for which (L,R) partitions Q.
>
> 2.  Ditto but with the reals defined instead to be intervals for which
> Q - (L U R) is a finite set.  ("Finite set" rather than just "finite" to
> avoid the other meaning of "finite interval."  The order plays no role
> in this definition, maximality of reals in the order being instead a
> theorem.)
>
> 3.  As for 2 but with "finite" replaced by "cardinality at most 1".
> The predicate "rational" is identified with the cardinality of Q - (L U R).
>
No constructivist (of whatever stripe) would be happy talking about
Q - (L U R), as in your second and third definitions, since he wouldn't
want to assume that L and R were complemented as subsets of Q.
Your first definition, if I understand it correctly, is equivalent to
what most toposophers would call the MacNeille reals -- that is, the
(Dedekind-MacNeille) order-completion of Q. If you "positivize" the
second and third (which would appear to be equivalent, for any sensible
notion of finiteness) by saying "Whenever p and q are rationals with
p < q, then either p \in L or q \in R", you get the Dedekind reals,
which are a proper subset of the MacNeille reals (though they
coincide iff De Morgan's law holds) but have rather better algebraic
properties.

Peter Johnstone