There's perhaps a confusion here with the "everyday" distinction between 'ordinal' and 'cardinal', on the one hand, and the strict mathematical definitions, on the other hand.
A cardinal number c may be defined mathematically as an "initial" ordinal, that is, an ordinal number c having the following property: no ordinal number b that is a strict subset of c can be order-equivalent to c. Under this definition, each cardinal number _is_ an ordinal number -- just a very special type of ordinal. On Sun, 18 Dec 2011, Raul Miller wrote: > Date: Sun, 18 Dec 2011 12:12:31 -0500 > From: Raul Miller<rauldmil...@gmail.com> > Subject: Re: [Jprogramming] J Midterm exam 2011 > To: Programming forum<programming@jsoftware.com> > Message-ID: > <cad2jou_tqbj4tuvp7y_wpgr9c7ywttpxn7kwqfajwgz_tcq...@mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > Yes. > > > That was a mistake. > > -- Raul On Sun, Dec 18, 2011 at 12:06 PM, Bo Jacoby <bojac...@yahoo.dk> > wrote: >> > Didn't you swap 'cardinal' and 'ordinal' below? >> > >> > I think that 'first' and 'second' are ordinal numbers while 'zero', 'one' >> > and 'two' are cardinal numbers. >> > >> > Ordinal numbers identify items, while cardinal numbers identify >> > in-betweens by counting sets of items. >> > >> > >> > This century is century no 21. The first year of this century is year no >> > 2001, and the last year of this century is year no 2100. Many people >> > celebrated the new century when year 1999 turned into year 2000 - one year >> > too early! Embarrassing! >> > >> > When this century started, 2000 years had passed. When this century ends, >> > 2100 years have passed. >> > >> > Arithmetic is done on cardinal numbers, not on ordinal numbers. That is >> > why i.3 is 0 1 2, I suppose. -- Murray Eisenberg mur...@math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
