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OT: Ordinal Numbers and Cardinal Numbers (was: Re: OFFLIST: Re: [Theory] Types of numerals)

From:Tom Chappell <tomhchappell@...>
Date:Tuesday, January 17, 2006, 15:53
Hello, Yahya.  Thanks for your reply.

--- Yahya Abdal-Aziz <yahya@...> wrote:
> -----Original Message----- >> [THC] I enjoyed your reply, and agreed with most of
>> [THC] and was informed by much of the rest. >> [THC] I say that to begin with because this reply
of mine
>> [THC] is going to concentrate on your only error. >> [THC] I didn't want you to think that was all I got
out of it.
> [YA] You're a good man, Tom H Chappell! :-)
[THC] Thanks.
> [YA] BTW, I love it that I'm usually wrong whenever > [YA] I make a bold statement - and people can see
> [YA] However, if I make a more cautious statement, > [YA] and hedge it with "it seems to me" or
> [YA] or "may", I think I manage to (unintentionally)
> [YA] bluff or confuse most readers, who then fail to
> [YA] call me on the error.
[THC] I hadn't noticed that, but it seems easy to believe.
> This is one reason I tend > to make assertions more boldly than my know- > ledge warrants; it's a great way of provoking > such necessary corrections as you have given me.
[THC] I usually say "if I understand correctly" or "as I [THC] understand it" or "as far as I know" or "I seem to recall" [THC] or some such.
> [YA] Simply saying "I'm not sure" doesn't have the > [YA] same effect ... So, while I have the greatest > [YA] respect for truth and accuracy, sometimes it > [YA] appears the only way to get at it is to
> [YA] by successive, more readily understandable, > [YA] approximations.
[THC] Seems reasonable. I'll try to remember that explanation.
> [YA] Thank you for your correction.
[THC] You're welcome. I'm glad it was something you felt [THC] thankful for.
> [YA] Would you publish the meat of it (without going
too far
> [YA] into the maths, eg models) on list?
[THC] I'd be glad too, but I don't know what to redact out, [THC] exactly. [THC] Could you send me the version you want me to publish [THC] on-list?
>> --- In, Yahya Abdal-Aziz >> <yahya@m...> wrote: >>> [snip] >>> On Fri, 13 Jan 2006 John Vertical wrote: >>>> [JV] (I do not know the precise set theoretical
>>>> [JV] of "ordinal", but I suspect it might deviate
from its
>>>> [JV] linguistic definition a little here.) >>> [YA] Set theory says nothing about order. You
>>> [YA] to go to algebra (specifically, the theory of >>> [YA] partial orders and lattice theory) to discuss >>> [YA] a "complete order" such as the ordinal series >>> [YA] first , second, third, ... corresponding to
>>> [YA] integer cardinals 1, 2, 3. >> [THC] This isn't so. Since my specialty is
>> [THC] Lattices, I am, of course, very intrigued by
>> [THC] treatment of Order; nevertheless "Ordinal
Numbers" is
>> [THC] well-treated in Set Theory. > [YA] I bow to your expertise.
[THC] Bow? That's probably more respect than I deserve! :-) [THC] If I ever finish my doctorate, maybe then I'll deserve it.
>> [THC] An "Ordinal Number" is the set of all
>> [THC] ordinal numbers. > [YA] Sounds exactly like my (hazy) recollections > [YA] of the definitions of cardinal numbers in
> [YA] maths at uni ...!
[THC] A cardinal can be defined as a certain kind of ordinal; [THC] namely, one that can't be put into one-to-one [THC] correspondence with any smaller ordinal.
> [YA] I could, of course, have confused > [YA] the two, having little use since then for the
> [YA] cardinal and ordinal beyond their everyday > [YA] connotations of "counting numbers" and "numbers > [YA] to identify list members by". >> [THC] The elements of an "Ordinal Number" are
>> [THC] by Membership. >> [THC] In other words, if W is an Ordinal Number,
and X and
>> [THC] Y and Z are elements (members) of W, >> [THC] 1. Either X is a member of Y, or Y is a
member of X,
>> [THC] or X=Y. >> [THC] 2. If X is a member of Y, and Y is a member
of Z,
>> [THC] then X is a member of Z. >> [THC] 3. If S is a nonempty subset of W, there is a
>> [THC] of S which is a member of every other member
of S.
>> [THC] Note: Every member of an Ordinal Number, is
also an
>> [THC] Ordinal Number. >> [THC] Note: If W is an Ordinal Number, then the
union of
>> [THC] W with {W} is an Ordinal Number also; for
short, it
>> [THC] is denoted W+1. >> [THC] Note: By the above, we can see that every
member of
>> [THC] any Ordinal Number, is also a subset of it.
That is,
>> [THC] if X is in Y and Y is in W, then X is in W;
so Y is
>> [THC] a subset of X. >> [THC] The first several Finite Ordinals are (using
{} to
>> [THC] mean "the Empty Set"; >> [THC] {} zero >> [THC] {{}} one = {zero} >> [THC] {{{}},{}} two = {one, zero} >> [THC] {{{{}},{}},{{}},{}} three = {two, one, zero} >> [THC] {{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}} >> [THC] four = {three, two, one, zero} >> [THC] And so on. > [YA] Yes, I do remember this series, but thought > [YA] them cardinals.
[THC] Actually every finite Ordinal is a finite Cardinal, if we [THC] adopt the above definition of "Cardinal"; none of them can [THC] be put into one-to-one correspondence with any smaller [THC] Ordinal.
> [YA] This much is probably worth > [YA] presenting onlist, don't you think?
[THC] Maybe, with an "OT" tag.
> [YA] But perhaps more simply, as : > [YA] zero = {} > [YA] one = {zero} > [YA] two = {zero, one} > [YA] three = {zero, one, two} > [YA] etc.
[THC] Do you really think people will get that [THC] one = {zero} = {{}} [THC] two = {one, zero} = {{{}}, {}} [THC] three = {two, one, zero} = {{{{}},{}}, {{}}, {}} [THC] four = {three, two, one, zero} [THC] = {{{{{}},{}},{{}},{}}, {{{}},{}}, {{}}, {}} [THC] from the above? [THC] Maybe they can.
>> [THC] The set of all Finite Ordinals is called
>> [THC] omega" (or "little omega"), which I
approximate as "w",
>> [THC] and is the smallest Infinite Ordinal. (For
some reason,
>> [THC] when discussing Ordinals, the favored word >> [THC] is "Transfinite" instead of "Infinite".) > [YA] Never did get a good explanation of > [YA] the distinction.
[THC] Neither have I. [THC] This URL answers the question: [THC] [THC] basically by saying, "infinite" can have more than one [THC] meaning, one of which is "transfinite". [THC] This URL: [THC] [THC] (which says, "transfinite number A.K.A. infinite number") [THC] Other URLs that discuss it are: [THC] [THC] [THC]
>> [THC] Any Ordinal that can be put into one-to-one >> [THC] correspondence with "w" is "Countable". > [YA] This is not necessarily too complex an idea > [YA] for a conlang. But from here on in, I suspect
> [YA] a very mathematical conculture would have need > [YA] of the number theory terms you discuss. What > [YA] do you think?
[THC] I'm not sure any natlang really needs words for "infinite". [THC] But the ideas of "discrete" vs "continuous", [THC] or "connected" vs "disconnected", seem natural.
>> [THC] The set of all Countable Ordinals is called >> [THC] "Uppercase Omega" (or "Big Omega"); >> [THC] I'll approximate it as _O_. >> [THC] That's what I was trying to tell John
>> [THC] "Ordinality" is a more primitive concept in
>> [THC] Theory than "Cardinality"; also, "Ordinality" >> [THC] is an intrinsic property, whereas
>> [THC] is an extrensic property. > [YA] Rings a vague bell ... But in natlangs, > [YA] don't we always learn to count > [YA] before we learn to group (form sets)?
[THC] Maybe not. I don't know, but I doubt the above.
> [YA} and don't we always learn to count > [YA} before we learn to sequence (order objects)?
[THC] Maybe so. I don't know, but I would guess that way. Tom H.C. in MI __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around