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 _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
