Cantor Confusion
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From: Dik T. Winter on 31 Oct 2006 19:46 In article <virgil-7942CE.14363831102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > In article <1162298882.038772.64700(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: .... > > > French mathematical usage, particularly Bourbaki, uses "positive" > > > and "strictly positive" where the correspponding English mathematical > > > usage is, respectively, "non-negative" and "positive". Similarly for > > > "negative' and "strictly negative" versus "non-positive" and "negative. > > > > Thanks, it is the same in German. > > WM being ambiguous again. > > Is the German usage the same as English or the same as French? It is the same as in English. Dutch usage was the French usage but that has been gradually replaced by the English usage. When I was at university, in some courses French usage was common, in other courses English usage. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 31 Oct 2006 20:05 In article <virgil-DF0CDC.14525831102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > In article <1162299524.423928.41670(a)k70g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > > The definition is sufficiently special (= non-general) to determine > > > > that lim [n-->oo] {1,2,3,...,n} = N is correct. > > > > > > Yes, of course, because that was the definition you provided. So because > > > it is so defined, the definition is correct. > > > > > > > > > > > The operator "lim [n-->oo]" defines N. In your example lim {n -> oo} > > > > {-1, 0, 1, ..., n} we have N too but in addition the numbers -1 and 0. > > > > > > "lim {n -> oo}" is an operator that works on sequences, apparently. And > > > you have defined that operator for precisely one sequence of sets. > > > > It is defined for infinitely many sets of integers. > > > > lim [n-->oo] {-k,-k+1,..., 0, 1,2,3,...,n} = {-k,-k+1, ...,0} u N for > > every k e N. > > lim [n-->oo] {k, k+1, k+2,...,n} = N \ {1,2,3,...,k-1} for every k e N. > > These are, at best, operational definitions which hold in particular > cases but do not define any general rule. > > Given, say, f:N -> P(N), a function from N to the power set of N, > Under what conditions on f can one say that there is an S in P(N) such > that lim[n -> oo] f(n) = S ? I once proposed a definition: Let f(n) be a function with domain N and as range subsets from some set Y. Now define: lim{n -> oo} f(n) = S exists if for each p in Y there is an n0 such that either p in f(n) for n >= n0, or p not in f(n) for n >= n0. And p in S if p in f(n) for n >= n0, and p not in S if p not in f(n) for n >= n0. Seems to me to be pretty straightforward and reasonable. With this definition we have the two limits Wolfgang insists on. Also it gives intuitive results like: lim{n -> oo} {n, n+1, ...} = {} But Wolfgang rejects it because it also gives: lim{n -> oo} {n+1, ..., 10n} = {}. And it may be noted that under this definition: lim{n -> oo} |f(n)| != |lim{n -> oo} f(n)| in general, even when both limits exist. (But in mathematics it is pretty common that you can not arbitrarily interchange two such operations.) The most obvious counter example: lim{n -> oo} {n} = {} but lim{n -> oo} 1 != 0. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 31 Oct 2006 20:50 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > All entries of the list have a finite number of letters. > > > > Correct. And given any integer, we can take a set of lines such that > > the number of letters in this set is greater than the integer. > > Correct. And given any set of lines we can find a finite number which > is larger. Otherwise, at least one of the lines would contain a number > of letters which was larger than any natural number, i.e. which was > omega. But what is important is not the number of letters in any given line but the number of lines. These are two different things. > > > > > An infinite > > > sequence is larger than any finite sequence. The diagonal of a list > > > cannot have more letters than the lines. > > > > > > > Correct. The number of letters in the lines is greater than any > > integer. > > Incorrect. Why do you say incorrect? Given any integer N, we certainly have more letters than N. > > Greater than any integer is no integer but only omega (an > its successors). > Call the number of letters anything you want. If it exists it must be greater than any integer. If it does not exist you cannot use it to limit the diagonal. > > So the greatest number of letters the diagonal can have is > > greater than any integer. > > Incorrect. See above. Your claim is that the number of letters the diagonal can have is limited above by the number of letters in the list. There are two possiblities: The number of letters in the list exists. In this case the number of letters in the list (whatever you call it) is greater than any integer. Thus the length of the diagonal is not limited to any integer. The number of letters in the list does not exist. In this case we cannot use this to limit the length of the diagonal. > > > > So the diagonal has infinite length (call this potentially infinite > > length if > > you get your kicks by saying potentially). > > The diagonal may have potentially infinite length, but that is less > than actual infinity, i.e., omega which is the first number larger than > any natural number. > So the diagonal has potentially infinite length. This means that given any integer N, the length of the diagonal is greater than N. However, this is less than omega which is the first number larger than every N. Where does the length of the diagonal fit? - William Hughes
From: stephen on 31 Oct 2006 22:10 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: >> In article <bdc92$45476e9e$82a1e228$30478(a)news1.tudelft.nl>, >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >> > It's easy to come up with a correct physical problem and >> > solve it with the wrong mathematics. As you did. >> >> HdB has come up with a mathematical problem which he claims can end the >> world: >> >> HdB claimed that a discontinuity in a mathematical function of time >> causes time to stop. And he claimed this followed from physics. >> >> Thus, if HdB is right, the vase problem will cause the end of the >> world. > Yes, but when? Noon? Stephen
From: Virgil on 31 Oct 2006 23:31
In article <ei937d$jsr$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > Virgil wrote: > >> In article <bdc92$45476e9e$82a1e228$30478(a)news1.tudelft.nl>, > >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> > >> > It's easy to come up with a correct physical problem and > >> > solve it with the wrong mathematics. As you did. > >> > >> HdB has come up with a mathematical problem which he claims can end the > >> world: > >> > >> HdB claimed that a discontinuity in a mathematical function of time > >> causes time to stop. And he claimed this followed from physics. > >> > >> Thus, if HdB is right, the vase problem will cause the end of the > >> world. > > > Yes, but when? > > Noon? > > Stephen As there cannot be any such noon according to HdB's physics experiments, it must happen before noon, but after every other time before noon. |