Re: Proportions of Infinity

From: Bruno Marchal <>
Date: Tue, 20 May 2003 16:54:56 +0200

At 10:10 20/05/03 +1000, Russell Standish wrote:

> > I agree with your "Meanings have measure, descriptions do not.", but
> for the
> > opposite reason: descriptions are countable, and meanings (models) are
> > uncountable, and so can be integrated upon.
> >
> > Bruno
> >
>Surely you don't mean this! The cardinality of the set of all
>descriptions is c. The dovetailer does enumerate them, but takes c
>timesteps to do so. In countable time, the dovetailer can only manage
>the set of finite prefixes.
>The set of meanings is countable. This is easy to see in the case of
>human literature, or language, but even in the case of biological
>phenotypes which appear to be characterised by continuous quantities,
>there comes a point where the difference between to phenotypic
>characteristics is too small to count as a difference. If anything,
>your own COMP hypothesis demands this to be the case.
>To see the mathematical picture, consider a countable partition of the
>interval [0,1]. The individual points of the interval have measure
>zero, but the subsets making up the partition may have positive measure.

I'm afraid we face the Logician/Physicist misunderstanding. I have
witness it more than one time before.
Physicist, like children, used the word model in the sense of the toys, the
"simplification" you can handle, and then tend to interpret their theories
as the real things. Cf. Children's reduced models as toys, and Bohr's model
of the atom.
Logicians, like painters, used the word model in the sense of the (intended)
real thing (generally infinite), like the standard model of Peano
Arithmetic, or ... the
model as the "real" nude women or guy in front of the painters, or sculptors.

(Just remember that physicists and logicians use the words "theory" and "model"
in opposite sense! Simplifying things just a little bit!).

I'm used to interpret the words "name" or "description" as intended for
finite things, like
programs or machines, or well presented formal theories. In computability
the models
(object of semantics, that is one of the logician's approach to the
"meanings") are the
"real" things, for example the description can be <a program for factorial
written in fortran> and
the real thing will be the infinite set of couples {(0,1) (1,1) (2,2)
(3,6), (4,24), (5,120), ...}.

Note that the UD generates both the descriptions and their "meanings",
because it generates
all programs, but also it executes them all. Of course "meanings" in the
sense occur when machines are able to project or anticipated their local
histories made
possible by continuum sheaves of similar histories. The section of that
sheaf has cardinality c.

Received on Tue May 20 2003 - 10:57:15 PDT

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