🚀 go-pugleaf

In article <1106@its63b.ed.ac.uk> jcb@lfcs.ed.ac.uk (J Bradfield) writes:
>In article <25739@cca.CCA.COM> g-rh@CCA.CCA.COM.UUCP (Richard Harter) writes:

	... Loose description of ordinals deleted

>If we're going to be pedantic and mathematical, let's at least get it right.
>The ordinals start at zero: ordinals are defined as equivalence classes of
>well-ordered sets, and the smallest ordinal is the order type of the empty
>set, to wit, zero. Cardinals are defined either as equivalence classes of
>ordinals (all those having the same size), or as the smallest ordinal in
>the equivalence class. For example, the smallest infinite cardinal,
>aleph-0, is the size of the sequence <0,1,2,...> -- but it's also the size
>of the sequence <0,1,2,...,0,1,2,...>, which has twice the length of the 
>first one. See any book on elementary set theory for more about ordinals
>and cardinals.

	Er, well, no.  Ordinals may be defined as equivalence classes of
well-ordered sets in set theories which permit this definition.  In set
theories which do not, e.g. Zermelo Frankel, they are defined quite
differently.  You are correct that most set theoretic definitions permit
an ordinal zero.  Transfinite arithmetic may be relevant for usenet 
discussions, but not programming languages :-).   The set theoretic
definitions, in any case, are not definitions of ordinals, per se;
they are definitions of sets that may be used to instantiate ordinals.
There are some fine logical points here that are blissfully ignored in
books on elementary set theory.

>All this is rather irrelevant to arrays; set theory is not necessarily
>a good guide for language designers!

	True enough.  The reason for mentioning ordinals is rather
more basic -- people use ordinals all of the time.  The problem is
really the more simple one that the first element of an array is at
displacement zero from the address of the array.  Some people conclude
from this that counting should start from zero.

>Personally, I use whichever base seems more appropriate for the task
>in hand---which is why I object to languages that won't let me start
>arrays at zero.

	Since it is (or so it seems to me) that one usually, in programming,
want to use indices as displacements, I agree, and would further say 
that that it is far more useful to have the default start at zero.
The problem is that it is rather more natural to think of loop indices
as ordinals, i.e. process the first instance, then the second, and so
on.  Experience says that it is simpler to treat them as displacemnt
values.
-- 

In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.
	Richard Harter, SMDS  Inc.