An uncountable countable set
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From: Dik T. Winter on 21 Sep 2006 09:13 In article <eet4ru$6ku$1(a)news.msu.edu> stephen(a)nomail.com writes: > cbrown(a)cbrownsystems.com 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. > > > Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an > > infinitesimal. Then what you are saying means > > > 1/B = 1/B > > 1*1/B = 1/B > > (B*1/B)*1/B = 1/B > > B*(1/B*1/B) = 1/B > > B*(1/B^2) = 1/B > > > Since 1/B is infinitesimal, its square is not distinct from 0; so... > > > B*(0) = 1/B > > 0 = 1/B > > > So 1/B is identical to 0. Where is my error? > > Assuming that Tony's definition of infinitesimal has anything > to do with any standard definition of infinitesimal. :) 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. -- 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 10:06 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Han de Bruijn wrote: >> I can't find any document on VM's site where he says that sqrt(2), pi, e >> and all the irrationals are not numbers. Would you mind to point me to a >> reference? (I've tried "On the abundance of the irrational numbers") > Oops! He _says_ it, in the document "Physical Constraints of Numbers": >> Irrational numbers simply are not numbers. > Weird ... > Han de Bruijn What is weird about it? It seems consistent with his position that infinite entities do not exist. I guess 'Representation is number. There is no difference. Numerals have no "soul".' was not as lucid as you thought, given that you did not understand what he meant. Perhaps this is why mathematicians prefer axioms and clear definitions. Are you going to apologize for the "babbling" remark? :) Stephen
From: stephen on 21 Sep 2006 10:27 Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > In article <eet4ru$6ku$1(a)news.msu.edu> stephen(a)nomail.com writes: > > cbrown(a)cbrownsystems.com 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. > > > > > Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an > > > infinitesimal. Then what you are saying means > > > > > 1/B = 1/B > > > 1*1/B = 1/B > > > (B*1/B)*1/B = 1/B > > > B*(1/B*1/B) = 1/B > > > B*(1/B^2) = 1/B > > > > > Since 1/B is infinitesimal, its square is not distinct from 0; so... > > > > > B*(0) = 1/B > > > 0 = 1/B > > > > > So 1/B is identical to 0. Where is my error? > > > > Assuming that Tony's definition of infinitesimal has anything > > to do with any standard definition of infinitesimal. :) > 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. I do not think Tony's infinitesimals are nil-potent. 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. I am not very familiar with any of the rigorous definitions of 'infinitesimal'. Stephen
From: Han de Bruijn on 21 Sep 2006 10:45 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>Han de Bruijn wrote: > >>>I can't find any document on VM's site where he says that sqrt(2), pi, e >>>and all the irrationals are not numbers. Would you mind to point me to a >>>reference? (I've tried "On the abundance of the irrational numbers") > >>Oops! He _says_ it, in the document "Physical Constraints of Numbers": > >>>Irrational numbers simply are not numbers. > >>Weird ... > > What is weird about it? It seems consistent with > his position that infinite entities do not exist. I guess > 'Representation is number. There is no difference. > Numerals have no "soul".' > was not as lucid as you thought, given that you did not > understand what he meant. Perhaps this is why mathematicians > prefer axioms and clear definitions. > > Are you going to apologize for the "babbling" remark? :) Herewith I do. Han de Bruijn
From: Tony Orlow on 21 Sep 2006 13:09
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? > > Brian Chandler > http://imaginatorium.org > Because N is defined as including all finite naturals, it is potentially infinite, each element having a finite index in the set. |