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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:

Hi,

> > > 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
inconsistencies.

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
sucessor,...

Moreover, stepping through a set might need other (efficiency, logical)
structures, you might have a look at Don Knuth prefascicles of his TAOCP
http://www-cs-faculty.stanford.edu/~knuth/taocp.html 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
nextItem()

> > 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
nextStep...

ciao
BF.

% PD Dr Bertfried Fauser
%     Institution: Max Planck Institute for Math, Leipzig
<http://www.mis.mpg.de>
%   Privat Docent: University of Konstanz, Phys Dept
<http://www.uni-konstanz.de>
%  contact|->URL : http://clifford.physik.uni-konstanz.de/~fauser/
%          Phone : Leipzig +49 341 9959 735  Konstanz +49 7531 693491

```