Hello, Yahya. Thanks for your reply.
--- Yahya Abdal-Aziz <yahya@...> 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 conlang@yahoogroups.com, 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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