An uncountable countable set
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From: Virgil on 21 Sep 2006 14:18 In article <4fd34$45123f60$82a1e228$7325(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > Assuming that Tony's definition of infinitesimal has anything > > to do with any standard definition of infinitesimal. :) > > What is "any standard definition of infinitesimal" ? Any of these in "non-standard analysis" as initiated by Abraham Robinson, or "hyperreal numbers".
From: Virgil on 21 Sep 2006 14:22 In article <4512c75c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Because N is defined as including all finite naturals, it is potentially > infinite, each element having a finite index in the set. If N is only potential then there is no actual, since N is as actual as any set.
From: imaginatorium on 21 Sep 2006 14:41 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> >> Actually infinite(S) <-> E seS A neN index(s)>n > >>>> >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > >>> So this is what you have now ('w' stands for 'N', which stands for?): > >> Where do you see 'w'? > >> > >>> S is actually infinite <-> Es(seS & An(new -> index(s)>n)) > >> I would rather not use 'w'. Sick to N. Why not? > >> > >>> S is potentially infinite <-> As(seS -> (index(s)ew & An(new -> Es(seS > >>> & index(s) = n)))) > >>> > >>> Okay, now I see that your 'actually infinite' is something like what > >>> set theory would describe as 'S has a member greater than any member of > >>> w'; and your 'potentially infinite' is what set theory would describe > >>> as 'S has only finite members but S has a denumerable number of finite > >>> members'. > >> Okay. > > > > Never mind the confusion over the letters - there seems to be a big > > problem here, in that the definition of "potentially-Tinfinite" and > > "actually-Tinfinite" makes reference to a set N, which I suppose is > > some sort of large set including lots (all?) natural numbers, of one > > flavour or another. How then would we ask whether this set N is either > > sort of Tinfinite? > Because N is defined as including all finite naturals, it is potentially > infinite, each element having a finite index in the set. It's all total mystery to me. In Orlovia, as I understand it, there's a sequence of finite naturals, that increase and increase and increase and somehow, under cover of a cloudy veil, transmute into "infinite naturals", which are "infinitely long" somehow, yet have two ends, because an "infinite natural" has the top bit set (is that right?) and yet this is all defined before we establish this pot-Tinfinite and act-Tinfinite stuff... But never mind, here's another thing that strikes me as extremely curious. Please say where I leave the (Orlovian) rails (if I do): Let Tn be the set of all finite naturals. Tn is a pot-Tinfinite set. Every finite natural in Tn has a finite index. Now consider Tomega (Tom for short), which is an Orlovian infinite number - of some sort; perhaps you can help out with details. Let Un be the union Tn U Tom. A set that includes every finite Tnatural, plus one other element. Presumably this set is now act-Tinfinite? That seems very strange, that just one element would make the difference between being pot- and act- vis a vis Tinfinitude? What about the set Tom U Tn. I forget whether the Tnats start at 1 or 0 (though I seem to recall you get emotional about it) - let's suppose it's 1. Then if we consider the following two sets: { 0, 1, 2, 3, ...[rest of the Tnats] } ...(1) pot-Tinfinite { Tom, 1, 2, 3, ...[rest of the Tnats] } ...(2) act-Tinfinite Then somehow replacing 0 in set (1) by Tom in set (2) changes a pot- to and act-. Can this be right? And what about a set like { [rest of the negative Tints] ... -4, -3, -2, -1, 0, 1, 2, 3, ...[rest of the Tnats] } ... (3) Set (3) looks Twice [spot the 'T'] as big (roughly) as set (2), yet is only pot-Tinfinite? Is that right? I breathlessly await, Sir, your response. Brian Chandler http://imaginatorium.org
From: Dik T. Winter on 21 Sep 2006 21:28 In article <eeu7fn$ti$2(a)news.msu.edu> stephen(a)nomail.com writes: > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > > > > Tony Orlow wrote: > > > >> I did give another curve with the same "Tlimit" as the staircase > > > >> in the limit, which produced an interesting result, giving weight > > > >> to the notion that an infinitesimal is something distinct from 0, > > > >> whose square is not distinct from 0. Note what Tony writes above. > > It has with one of the definitions. Have a look at "synthetic differential > > geometry". Lavendhomme, Kock, amongst others. They use infinitesimals > > that are nil-potent. But, of course, they deny the law of the excluded > > middle in that system. > > From the context, I gather that nil-potent infinitesimals > are infinitesimals whose square equals 0? I could not find > a clear definition on the web, but that interpretation seems > consistent with what I did find. That is it. I believe Anders Kock has quite some information online. > I do not think Tony's infinitesimals are nil-potent. But see what Tony did write! > Tony's > infinitesimals are just really small numbers whose inverses are > infinite that otherwise behave just like ordinary real numbers. > I think that Tony's idea of infinitesimals is actually more > in line with Robision's infinitesimals which do not have > nilpotent infinitesimals, if my understanding is correct. Indeed, Robinson's infinitesimals are not nil-potent. But if Tony states that his infinitesimals are nil-potent, they are closer to the infinitesimals of Kock. Off-hand, I do not remember whether inversion of these infinitesimals does work. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: stephen on 21 Sep 2006 22:15
Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > In article <eeu7fn$ti$2(a)news.msu.edu> stephen(a)nomail.com writes: > > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > > > > > Tony Orlow wrote: > > > > >> I did give another curve with the same "Tlimit" as the staircase > > > > >> in the limit, which produced an interesting result, giving weight > > > > >> to the notion that an infinitesimal is something distinct from 0, > > > > >> whose square is not distinct from 0. > Note what Tony writes above. I missed that. > > > It has with one of the definitions. Have a look at "synthetic differential > > > geometry". Lavendhomme, Kock, amongst others. They use infinitesimals > > > that are nil-potent. But, of course, they deny the law of the excluded > > > middle in that system. > > > > From the context, I gather that nil-potent infinitesimals > > are infinitesimals whose square equals 0? I could not find > > a clear definition on the web, but that interpretation seems > > consistent with what I did find. > That is it. I believe Anders Kock has quite some information online. Thanks. > > I do not think Tony's infinitesimals are nil-potent. > But see what Tony did write! Yes. He has apparently changed his mind, or the properties of his infinitesimals depend on his current argument. > > Tony's > > infinitesimals are just really small numbers whose inverses are > > infinite that otherwise behave just like ordinary real numbers. > > I think that Tony's idea of infinitesimals is actually more > > in line with Robision's infinitesimals which do not have > > nilpotent infinitesimals, if my understanding is correct. > Indeed, Robinson's infinitesimals are not nil-potent. But if Tony > states that his infinitesimals are nil-potent, they are closer to > the infinitesimals of Kock. Off-hand, I do not remember whether > inversion of these infinitesimals does work. Well as I said, I doubt Tony's infinitesimals really correspond to any of the standard definitions. Stephen |