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Re: [Axiom-developer] StepThrough

From: Bertfried Fauser
Subject: Re: [Axiom-developer] StepThrough
Date: Fri, 4 Nov 2005 19:01:40 +0100 (CET)

On Fri, 4 Nov 2005, C Y wrote:


> > > I am not exactly sure what you mean by *model* in this case but I
> > do not
> > > think Float is any more of a model for reals than is Fraction
> > Integer.
> > > It is no more difficult to define 'nextItem' in Float than it is in
> > Fraction
> > > Integer. Instead of 'numer' and 'denom' we have 'mantissa' and
> > 'exponent'.
> >
> > OK, I surrender.
> So we're agreeing nextItem makes sense in Float?

NO! IFF float is a model for 'floats' then such a domain/category should
NOT have an attribute COUNTABLE, otherwise it may run into logical

I promote this opinion even if I know that an actual digital computer has
only a finite (not even countable!) number of such objects available at a
moment, but one can choose at random from an uncountable resource and a
successor function (NextItem) does not make sense.

I am not even sure if I appreciate that one has a 'standatrd?' nextItem in
the rationals. It is quite not clear, what a unique (canonical) order of
teh rationals is. Stepping through a finite or countable set means to
implement a total order starting with the smales element init() and ending
with the greatest (if finite). (Streams are infinite such things and I do
not really see why one should not even have a sort of 'stream' object if a
category has countable, something like nextItem() is nothing but a stream
(isn't it?) However a set may have many total orderings which can be
different, hence its not canonical.

Eg what are the nextItems of partitons, compositions, etc. This is a
concept which depends on an order. So first we would need to introduce an
(total) order then there is a canonical nextItem.

Partitions can be partially ordered, so that you do not find a unique

Moreover, stepping through a set might need other (efficiency, logical)
structures, you might have a look at Don Knuth prefascicles of his TAOCP where eg codes are
discussed so that you step through binary strings and in each step exactly
one bit changes and other such options. Any such option needs a futher

> > Since the set of computable numbers is countable and we can clearly
> > only define domains containing computable numbers in Axiom, all
> > domains would have COUNTABLE. Of course for some domains it will be
> > more difficult to come up with an enumeration than for others.
> Indeed.

You can algebraically define %pi and %e, of course you cannot give a
digital or decimal presentations, but would that be desirable? %pi is fine
for me ;-)

Anyway, this mail shall not prevent you from starting doing something with


% PD Dr Bertfried Fauser
%     Institution: Max Planck Institute for Math, Leipzig 
%   Privat Docent: University of Konstanz, Phys Dept 
%  contact|->URL :
%          Phone : Leipzig +49 341 9959 735  Konstanz +49 7531 693491

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