From: Andrej Bauer <Andrej.Bauer@andrej.com>
To: categories@mta.ca
Subject: Re: Cauchy completeness of Cauchy reals
Date: 27 Jan 2003 18:41:39 +0100 [thread overview]
Message-ID: <vkafzrepk18.fsf@laurie.fmf.uni-lj.si> (raw)
In-Reply-To: Alex Simpson's message of "Mon, 27 Jan 2003 03:57:45 +0000 (GMT)"
Alex Simpson <als@inf.ed.ac.uk> writes:
> In fact not. Markov's principle holds in the effective topos, and
> there, unless I'm much mistaken, it is not even true that the map from
> binary representations to Cauchy (= Dedekind) reals in [0,1] is epi,
> let alone split epi.
The map e : 2^N --> [0,1] defined by
e x = x_0/2 + x_1/4 + x_2/8 + ...
is epi in the effective topos, but it is not regular epi. In terms of
logic, this means that
forall a : [0,1]. (not not (exists x : 2^n. (e x = a)))
is valid in the effective topos, but
forall a : [0,1]. (exists x : 2^n. (e x = a))
is not valid.
Andrej
next reply other threads:[~2003-01-27 17:41 UTC|newest]
Thread overview: 21+ messages / expand[flat|nested] mbox.gz Atom feed top
2003-01-27 17:41 Andrej Bauer [this message]
2003-01-28 1:50 ` Alex Simpson
-- strict thread matches above, loose matches on Subject: below --
2003-01-28 20:51 Dusko Pavlovic
2003-01-29 2:00 ` Toby Bartels
2003-01-29 8:35 ` Alex Simpson
2003-02-04 9:15 ` Dusko Pavlovic
2003-02-05 20:56 ` Toby Bartels
2003-01-28 9:44 Andrej Bauer
2003-01-16 23:05 Generalization of Browder's F.P. Theorem? Michael Barr
2003-01-21 18:11 ` Andrej Bauer
2003-01-22 10:13 ` Cauchy completeness of Cauchy reals Martin Escardo
2003-01-22 23:33 ` Dusko Pavlovic
2003-01-24 8:51 ` Martin Escardo
2003-01-25 2:21 ` Dusko Pavlovic
2003-01-25 16:24 ` Prof. Peter Johnstone
2003-01-27 3:57 ` Alex Simpson
2003-01-23 6:29 ` Vaughan Pratt
2003-02-04 0:47 ` Vaughan Pratt
2003-02-05 16:06 ` Prof. Peter Johnstone
2003-01-23 9:50 ` Mamuka Jibladze
2003-01-24 1:34 ` Ross Street
2003-01-24 16:56 ` Dusko Pavlovic
2003-01-24 19:48 ` Dusko Pavlovic
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