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CONLANG  January 2006, Week 3

CONLANG January 2006, Week 3

Subject:

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

From:

Tom Chappell <[log in to unmask]>

Reply-To:

Constructed Languages List <[log in to unmask]>

Date:

Tue, 17 Jan 2006 07:53:30 -0800

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (237 lines)

Hello, Yahya.  Thanks for your reply.

--- Yahya Abdal-Aziz <[log in to unmask]> wrote:
> -----Original Message-----
>> [THC] I enjoyed your reply, and agreed with most of
it;
>> [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
it.
> [YA] However, if I make a more cautious statement,
> [YA] and hedge it with "it seems to me" or
"probably"
> [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
approach
> [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 [log in to unmask], Yahya Abdal-Aziz
>> <yahya@m...> wrote:
>>> [snip]
>>> On Fri, 13 Jan 2006 John Vertical wrote:
>>>> [JV] (I do not know the precise set theoretical
definition 
>>>> [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
have
>>> [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
the
>>> [YA] integer cardinals 1, 2, 3.
>> [THC] This isn't so.  Since my specialty is
Multiplicative
>> [THC] Lattices, I am, of course, very intrigued by
Algebra's 
>> [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
previous
>> [THC] ordinal numbers.
> [YA] Sounds exactly like my (hazy) recollections 
> [YA] of the definitions of cardinal numbers in
third-year
> [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
terms 
> [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
well-ordered
>> [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
member
>> [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
"lowercase
>> [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]
http://mathforum.org/library/drmath/view/51849.html
[THC] basically by saying, "infinite" can have more
than one 
[THC] meaning, one of which is "transfinite".
[THC] This URL:
[THC] http://en.wikipedia.org/wiki/Transfinite_number
[THC] (which says, "transfinite number A.K.A. infinite
number")
[THC] Other URLs that discuss it are: 
[THC] http://24.62.177.166:8080//infin.htm
[THC] http://www.bartleby.com/65/tr/transfin.html
[THC]
http://www.c3.lanl.gov/mega-math/gloss/infinity/xfinite.html
>> [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
only 
> [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
Vertical.
>> [THC] "Ordinality" is a more primitive concept in
Set
>> [THC] Theory than "Cardinality"; also, "Ordinality"
>> [THC] is an intrinsic property, whereas
"Cardinality"
>> [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


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January 2008, Week 4
January 2008, Week 3
January 2008, Week 2
January 2008, Week 1
December 2007, Week 5
December 2007, Week 4
December 2007, Week 3
December 2007, Week 2
December 2007, Week 1
November 2007, Week 5
November 2007, Week 4
November 2007, Week 3
November 2007, Week 2
November 2007, Week 1
October 2007, Week 5
October 2007, Week 4
October 2007, Week 3
October 2007, Week 2
October 2007, Week 1
September 2007, Week 5
September 2007, Week 4
September 2007, Week 3
September 2007, Week 2
September 2007, Week 1
August 2007, Week 5
August 2007, Week 4
August 2007, Week 3
August 2007, Week 2
August 2007, Week 1
July 2007, Week 5
July 2007, Week 4
July 2007, Week 3
July 2007, Week 2
July 2007, Week 1
June 2007, Week 5
June 2007, Week 4
June 2007, Week 3
June 2007, Week 2
June 2007, Week 1
May 2007, Week 5
May 2007, Week 4
May 2007, Week 3
May 2007, Week 2
May 2007, Week 1
April 2007, Week 5
April 2007, Week 4
April 2007, Week 3
April 2007, Week 2
April 2007, Week 1
March 2007, Week 5
March 2007, Week 4
March 2007, Week 3
March 2007, Week 2
March 2007, Week 1
February 2007, Week 4
February 2007, Week 3
February 2007, Week 2
February 2007, Week 1
January 2007, Week 5
January 2007, Week 4
January 2007, Week 3
January 2007, Week 2
January 2007, Week 1
December 2006, Week 5
December 2006, Week 4
December 2006, Week 3
December 2006, Week 2
December 2006, Week 1
November 2006, Week 5
November 2006, Week 4
November 2006, Week 3
November 2006, Week 2
November 2006, Week 1
October 2006, Week 5
October 2006, Week 4
October 2006, Week 3
October 2006, Week 2
October 2006, Week 1
September 2006, Week 5
September 2006, Week 4
September 2006, Week 3
September 2006, Week 2
September 2006, Week 1
August 2006, Week 5
August 2006, Week 4
August 2006, Week 3
August 2006, Week 2
August 2006, Week 1
July 2006, Week 5
July 2006, Week 4
July 2006, Week 3
July 2006, Week 2
July 2006, Week 1
June 2006, Week 5
June 2006, Week 4
June 2006, Week 3
June 2006, Week 2
June 2006, Week 1
May 2006, Week 5
May 2006, Week 4
May 2006, Week 3
May 2006, Week 2
May 2006, Week 1
April 2006, Week 5
April 2006, Week 4
April 2006, Week 3
April 2006, Week 2
April 2006, Week 1
March 2006, Week 5
March 2006, Week 4
March 2006, Week 3
March 2006, Week 2
March 2006, Week 1
February 2006, Week 4
February 2006, Week 3
February 2006, Week 2
February 2006, Week 1
January 2006, Week 5
January 2006, Week 4
January 2006, Week 3
January 2006, Week 2
January 2006, Week 1
December 2005, Week 5
December 2005, Week 4
December 2005, Week 3
December 2005, Week 2
December 2005, Week 1
November 2005, Week 5
November 2005, Week 4
November 2005, Week 3
November 2005, Week 2
November 2005, Week 1
October 2005, Week 5
October 2005, Week 4
October 2005, Week 3
October 2005, Week 2
October 2005, Week 1
September 2005, Week 5
September 2005, Week 4
September 2005, Week 3
September 2005, Week 2
September 2005, Week 1
August 2005, Week 5
August 2005, Week 4
August 2005, Week 3
August 2005, Week 2
August 2005, Week 1
July 2005, Week 5
July 2005, Week 4
July 2005, Week 3
July 2005, Week 2
July 2005, Week 1
June 2005, Week 5
June 2005, Week 4
June 2005, Week 3
June 2005, Week 2
June 2005, Week 1
May 2005, Week 5
May 2005, Week 4
May 2005, Week 3
May 2005, Week 2
May 2005, Week 1
April 2005, Week 5
April 2005, Week 4
April 2005, Week 3
April 2005, Week 2
April 2005, Week 1
March 2005, Week 5
March 2005, Week 4
March 2005, Week 3
March 2005, Week 2
March 2005, Week 1
February 2005, Week 4
February 2005, Week 3
February 2005, Week 2
February 2005, Week 1
January 2005, Week 5
January 2005, Week 4
January 2005, Week 3
January 2005, Week 2
January 2005, Week 1
December 2004, Week 5
December 2004, Week 4
December 2004, Week 3
December 2004, Week 2
December 2004, Week 1
November 2004, Week 5
November 2004, Week 4
November 2004, Week 3
November 2004, Week 2
November 2004, Week 1
October 2004, Week 5
October 2004, Week 4
October 2004, Week 3
October 2004, Week 2
October 2004, Week 1
September 2004, Week 5
September 2004, Week 4
September 2004, Week 3
September 2004, Week 2
September 2004, Week 1
August 2004, Week 5
August 2004, Week 4
August 2004, Week 3
August 2004, Week 2
August 2004, Week 1
July 2004, Week 5
July 2004, Week 4
July 2004, Week 3
July 2004, Week 2
July 2004, Week 1
June 2004, Week 5
June 2004, Week 4
June 2004, Week 3
June 2004, Week 2
June 2004, Week 1
May 2004, Week 5
May 2004, Week 4
May 2004, Week 3
May 2004, Week 2
May 2004, Week 1
April 2004, Week 5
April 2004, Week 4
April 2004, Week 3
April 2004, Week 2
April 2004, Week 1
March 2004, Week 5
March 2004, Week 4
March 2004, Week 3
March 2004, Week 2
March 2004, Week 1
February 2004, Week 5
February 2004, Week 4
February 2004, Week 3
February 2004, Week 2
February 2004, Week 1
January 2004, Week 5
January 2004, Week 4
January 2004, Week 3
January 2004, Week 2
January 2004, Week 1
December 2003, Week 5
December 2003, Week 4
December 2003, Week 3
December 2003, Week 2
December 2003, Week 1
November 2003, Week 5
November 2003, Week 4
November 2003, Week 3
November 2003, Week 2
November 2003, Week 1
October 2003, Week 5
October 2003, Week 4
October 2003, Week 3
October 2003, Week 2
October 2003, Week 1
September 2003, Week 5
September 2003, Week 4
September 2003, Week 3
September 2003, Week 2
September 2003, Week 1
August 2003, Week 5
August 2003, Week 4
August 2003, Week 3
August 2003, Week 2
August 2003, Week 1
July 2003, Week 5
July 2003, Week 4
July 2003, Week 3
July 2003, Week 2
July 2003, Week 1
June 2003, Week 5
June 2003, Week 4
June 2003, Week 3
June 2003, Week 2
June 2003, Week 1
May 2003, Week 5
May 2003, Week 4
May 2003, Week 3
May 2003, Week 2
May 2003, Week 1
April 2003, Week 5
April 2003, Week 4
April 2003, Week 3
April 2003, Week 2
April 2003, Week 1
March 2003, Week 5
March 2003, Week 4
March 2003, Week 3
March 2003, Week 2
March 2003, Week 1
February 2003, Week 4
February 2003, Week 3
February 2003, Week 2
February 2003, Week 1
January 2003, Week 5
January 2003, Week 4
January 2003, Week 3
January 2003, Week 2
January 2003, Week 1
December 2002, Week 5
December 2002, Week 4
December 2002, Week 3
December 2002, Week 2
December 2002, Week 1
November 2002, Week 5
November 2002, Week 4
November 2002, Week 3
November 2002, Week 2
November 2002, Week 1
October 2002, Week 5
October 2002, Week 4
October 2002, Week 3
October 2002, Week 2
October 2002, Week 1
September 2002, Week 5
September 2002, Week 4
September 2002, Week 3
September 2002, Week 2
September 2002, Week 1
August 2002, Week 5
August 2002, Week 4
August 2002, Week 3
August 2002, Week 2
August 2002, Week 1
July 2002, Week 5
July 2002, Week 4
July 2002, Week 3
July 2002, Week 2
July 2002, Week 1
June 2002, Week 5
June 2002, Week 4
June 2002, Week 3
June 2002, Week 2
June 2002, Week 1
May 2002, Week 5
May 2002, Week 4
May 2002, Week 3
May 2002, Week 2
May 2002, Week 1
April 2002, Week 5
April 2002, Week 4
April 2002, Week 3
April 2002, Week 2
April 2002, Week 1
March 2002, Week 5
March 2002, Week 4
March 2002, Week 3
March 2002, Week 2
March 2002, Week 1
February 2002, Week 4
February 2002, Week 3
February 2002, Week 2
February 2002, Week 1
January 2002, Week 5
January 2002, Week 4
January 2002, Week 3
January 2002, Week 2
January 2002, Week 1
December 2001, Week 5
December 2001, Week 4
December 2001, Week 3
December 2001, Week 2
December 2001, Week 1
November 2001, Week 5
November 2001, Week 4
November 2001, Week 3
November 2001, Week 2
November 2001, Week 1
October 2001, Week 5
October 2001, Week 4
October 2001, Week 3
October 2001, Week 2
October 2001, Week 1
September 2001, Week 5
September 2001, Week 4
September 2001, Week 3
September 2001, Week 2
September 2001, Week 1
August 2001, Week 5
August 2001, Week 4
August 2001, Week 3
August 2001, Week 2
August 2001, Week 1
July 2001, Week 5
July 2001, Week 4
July 2001, Week 3
July 2001, Week 2
July 2001, Week 1
June 2001, Week 5
June 2001, Week 4
June 2001, Week 3
June 2001, Week 2
June 2001, Week 1
May 2001, Week 5
May 2001, Week 4
May 2001, Week 3
May 2001, Week 2
May 2001, Week 1
April 2001, Week 5
April 2001, Week 4
April 2001, Week 3
April 2001, Week 2
April 2001, Week 1
March 2001, Week 5
March 2001, Week 4
March 2001, Week 3
March 2001, Week 2
March 2001, Week 1
February 2001, Week 4
February 2001, Week 3
February 2001, Week 2
February 2001, Week 1
January 2001, Week 5
January 2001, Week 4
January 2001, Week 3
January 2001, Week 2
January 2001, Week 1
December 2000, Week 5
December 2000, Week 4
December 2000, Week 3
December 2000, Week 2
December 2000, Week 1
November 2000, Week 5
November 2000, Week 4
November 2000, Week 3
November 2000, Week 2
November 2000, Week 1
October 2000, Week 5
October 2000, Week 4
October 2000, Week 3
October 2000, Week 2
October 2000, Week 1
September 2000, Week 5
September 2000, Week 4
September 2000, Week 3
September 2000, Week 2
September 2000, Week 1
August 2000, Week 5
August 2000, Week 4
August 2000, Week 3
August 2000, Week 2
August 2000, Week 1
July 2000, Week 5
July 2000, Week 4
July 2000, Week 3
July 2000, Week 2
July 2000, Week 1
June 2000, Week 5
June 2000, Week 4
June 2000, Week 3
June 2000, Week 2
June 2000, Week 1
May 2000, Week 5
May 2000, Week 4
May 2000, Week 3
May 2000, Week 2
May 2000, Week 1
April 2000, Week 5
April 2000, Week 4
April 2000, Week 3
April 2000, Week 2
April 2000, Week 1
March 2000, Week 5
March 2000, Week 4
March 2000, Week 3
March 2000, Week 2
March 2000, Week 1
February 2000, Week 5
February 2000, Week 4
February 2000, Week 3
February 2000, Week 2
February 2000, Week 1
January 2000, Week 5
January 2000, Week 4
January 2000, Week 3
January 2000, Week 2
January 2000, Week 1
December 1999, Week 5
December 1999, Week 4
December 1999, Week 3
December 1999, Week 2
December 1999, Week 1
November 1999, Week 5
November 1999, Week 4
November 1999, Week 3
November 1999, Week 2
November 1999, Week 1
October 1999, Week 5
October 1999, Week 4
October 1999, Week 3
October 1999, Week 2
October 1999, Week 1
September 1999, Week 5
September 1999, Week 4
September 1999, Week 3
September 1999, Week 2
September 1999, Week 1
August 1999, Week 5
August 1999, Week 4
August 1999, Week 3
August 1999, Week 2
August 1999, Week 1
July 1999, Week 5
July 1999, Week 4
July 1999, Week 3
July 1999, Week 2
July 1999, Week 1
June 1999, Week 5
June 1999, Week 4
June 1999, Week 3
June 1999, Week 2
June 1999, Week 1
May 1999, Week 5
May 1999, Week 4
May 1999, Week 3
May 1999, Week 2
May 1999, Week 1
April 1999, Week 5
April 1999, Week 4
April 1999, Week 3
April 1999, Week 2
April 1999, Week 1
March 1999, Week 5
March 1999, Week 4
March 1999, Week 3
March 1999, Week 2
March 1999, Week 1
February 1999, Week 5
February 1999, Week 4
February 1999, Week 3
February 1999, Week 2
February 1999, Week 1
January 1999, Week 5
January 1999, Week 4
January 1999, Week 3
January 1999, Week 2
January 1999, Week 1
December 1998, Week 5
December 1998, Week 4
December 1998, Week 3
December 1998, Week 2
December 1998, Week 1
November 1998, Week 5
November 1998, Week 4
November 1998, Week 3
November 1998, Week 2
November 1998, Week 1
October 1998, Week 5
October 1998, Week 4
October 1998, Week 3
October 1998, Week 2
October 1998, Week 1
September 1998, Week 5
September 1998, Week 4
September 1998, Week 3

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