From: | Tom Chappell <tomhchappell@...> |
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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 ofit;>> [THC] and was informed by much of the rest. >> [THC] I say that to begin with because this replyof mine>> [THC] is going to concentrate on your only error. >> [THC] I didn't want you to think that was all I gotout 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 seeit.> [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 toapproach> [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 goingtoo 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 theoreticaldefinition>>>> [JV] of "ordinal", but I suspect it might deviatefrom its>>>> [JV] linguistic definition a little here.) >>> [YA] Set theory says nothing about order. Youhave>>> [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 tothe>>> [YA] integer cardinals 1, 2, 3. >> [THC] This isn't so. Since my specialty isMultiplicative>> [THC] Lattices, I am, of course, very intrigued byAlgebra's>> [THC] treatment of Order; nevertheless "OrdinalNumbers" 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 allprevious>> [THC] ordinal numbers. > [YA] Sounds exactly like my (hazy) recollections > [YA] of the definitions of cardinal numbers inthird-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 theterms> [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" arewell-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 amember of X,>> [THC] or X=Y. >> [THC] 2. If X is a member of Y, and Y is a memberof Z,>> [THC] then X is a member of Z. >> [THC] 3. If S is a nonempty subset of W, there is amember>> [THC] of S which is a member of every other memberof S.>> [THC] Note: Every member of an Ordinal Number, isalso an>> [THC] Ordinal Number. >> [THC] Note: If W is an Ordinal Number, then theunion of>> [THC] W with {W} is an Ordinal Number also; forshort, it>> [THC] is denoted W+1. >> [THC] Note: By the above, we can see that everymember 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 Iapproximate as "w",>> [THC] and is the smallest Infinite Ordinal. (Forsome 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 suspectonly> [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 JohnVertical.>> [THC] "Ordinality" is a more primitive concept inSet>> [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 __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com