Ah... true... But there's at least two different definitions in use for natural numbers. These correspond to APL's
[]IO <- 0 and []IO <- 1 See also: https://en.wikipedia.org/wiki/Natural_number Thanks, -- Raul On Thu, May 31, 2018 at 2:13 PM, Donna Y <[email protected]> wrote: > In any case it has a number system that includes natural numbers as a subset > and natural numbers are both cardinal and ordinal. > > Donna Y > [email protected] > > >> On May 31, 2018, at 1:50 PM, Raul Miller <[email protected]> wrote: >> >> J has complex numbers, including imaginary numbers, actually. >> >> Thanks, >> >> -- >> Raul >> >> >> On Thu, May 31, 2018 at 1:42 PM, Donna Y <[email protected]> wrote: >>> There are Natural numbers that can be used for counting (Cardinal) and >>> ordering (Ordinal). >>> >>> Indexing arrays is an instance of Ordinals. >>> >>> Counting elements in arrays is an instance of Cardinal. >>> >>> J might not have Irrational or Imaginary or Complex numbers but it does >>> have Natural numbers which can be used as Ordinal or Cardinal even if J >>> does not declare that type. There might be Real or Integer or Rational >>> numbers. The natural numbers with 0, correspond to the non-negative integers >>> >>> >>> Donna Y >>> [email protected] >>> >>> >>>> On May 31, 2018, at 1:22 PM, Raul Miller <[email protected]> wrote: >>>> >>>> On Thu, May 31, 2018 at 11:48 AM, Jose Mario Quintana >>>> <[email protected]> wrote: >>>>>> Are you referring to the notation you invented, here? >>>>> >>>>> The notation I invented? >>>> >>>> Oops, I thought you were Bo, for some reason. I don't remember all the >>>> details of the notations he has proposed. But that's my mistake and >>>> not a relevant tangent in this thread, for now at least. >>>> >>>>>> When I try to look up "finite mathematical ordinals" I don't see >>>>>> anything significant with that label. And when I try to parse that >>>>> >>>>> In general, mathematical ordinals and mathematical cardinals are not the >>>>> same. >>>> >>>> They are indeed different abstractions. Howeve, that does not mean >>>> that there's no equivalences between them. >>>> >>>>>> phrase as individual words, I see no contradiction with what I had >>>>>> said. >>>>> >>>>> I do not see one either (often I respond to posts in sequence without >>>>> necessarily having read all the subsequent posts). >>>> >>>> Fair enough. >>>> >>>> Thanks, >>>> >>>> -- >>>> Raul >>>> ---------------------------------------------------------------------- >>>> For information about J forums see http://www.jsoftware.com/forums.htm >>> >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
