From: Andrej Bauer <Andrej.Bauer@andrej.com>
To: categories@mta.ca
Subject: Re: arithmetical and geometric reals in (models of) SDG
Date: Wed, 07 Apr 2004 15:00:23 +0200 [thread overview]
Message-ID: <834qrw0wpk.fsf@haka.fmf.uni-lj.si> (raw)
In-Reply-To: <200403261407.PAA29318@fb04209.mathematik.tu-darmstadt.de>
Thomas Streicher <streicher@mathematik.tu-darmstadt.de> writes:
>
> Recently I was asking myself what is the relation between the arithmetical
> (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms
> (i.e. at least the Kock-Lawvere) axiom.
Perhaps it is worth mentioning that in the ring of smooth reals R the
sequence a_k = 2^(-k) is Cauchy but has many "limits" because every
infinitesimal dx satisfies the condition "dx is the limit of a_k".
This shows that R is not Cauchy complete, not because limits of Cauchy
sequences are missing but because there are too many.
I once thought the above observation implied there can be no isometric
embedding of a Cauchy-complete field (e.g. the Dedekind reals) into R,
but now I am not convinced anymore.
Andrej Bauer
Department of Mathematics and Physics
University of Ljubljana
http://andrej.com
prev parent reply other threads:[~2004-04-07 13:00 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2004-03-26 14:07 Thomas Streicher
2004-04-07 13:00 ` Andrej Bauer [this message]
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