Cantor Confusion
in [Math]
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From: WM on 17 May 2007 14:51 On 17 Mai, 02:24, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1179346353.430169.24...(a)y80g2000hsf.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 16 Mai, 04:01, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > The set is finitely defined. Not all lucky numbers can get finitely > > > > defined. > > > > > > In general a recursive definition is not considered a finite definition. > > > > Every definition which ends after finitely many words is a finite > > definition. > > Ah, so you disagree with common mathematical terminology. No. A finite definition means a definition by a finite number of words. Every other definition is nonsense. I agree with the common mathematical definition which implies that there are only finitely many definitions. Regards, WM
From: MoeBlee on 17 May 2007 15:27 On May 17, 11:51 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > No. A finite definition means a definition by a finite number of > words. Every other definition is nonsense. I agree with the common > mathematical definition which implies that there are only finitely > many definitions. I always love seeing each day the latest nonsense written by WM. For example: "The common mathematical definition which implies that there are only finitely many definitions". Now that just tickles me with amusement! MoeBlee
From: WM on 17 May 2007 16:23 On 17 Mai, 02:23, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1179346082.257799.291...(a)k79g2000hse.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 16 Mai, 03:55, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1178795907.957410.94...(a)o5g2000hsb.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > > > On 10 Mai, 03:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > So in your opinion there are only finitely many paths in the > > > > > > > infinite tree. > ... > > > > That is correct. But that does not mean that I said that there are > > > > only finitely many paths in the tree, as you implied. > > > > > > Why not? I can ask only for finitely many paths, and (if I interprete your > > > thinking correctly) path which I can not ask for do not exist. > > > > That is MatheRealism. > > See the "in your opinion". And you also want to apply that rule to > mathematics. In general I do not apply MatheRealism when refuting set theory (because on that basis there are no infinite sets to refute). > > > > > It is valid for those cases I did prove. I did not prove it (and one > > > > cannot prove it) for "all natural numbers which obey this formula", > > > > because all natural numbers which obey this formula do not obey this > > > > formula. > > > > > > This sentence makes no sense. I do not ask it for "all natural numbers > > > which obey this formula", I ask it for "all natural numbers". And what > > > natural numbers that obey that formula do not obey that formula? > > > > Up to each one they obey it. *All* natural numbers obviously do not > > obey the formula. > > What do you mean with the "upto"? The formula holds for each natural number > n, and so it holds for all natural numbers n. Consider the formula again: > sum{i = 1..n} i = (n + 1) * n / 2. > I state it holds for each natural number n (there is no upto involved in > the statement), and so it holds for all natural numbers n. That is *not* > a statement about: > sum{i = 1..oo} > because in that case we have not substituted a natural number for n. > If something is stated to hold for all natural numbers, that means that > it holds for each and every natural number, *not* that it holds for the > set of natural numbers. > If a sum is stated to hold for every subset of natural numbers then it has to hold for the whole set too. The same is true for initial segments. Therefore we can put the sum over all natural numbers on the left hand side. > > > > My argument is: Why do you have to exclude the case 0.999... = > > > > 1.000... if you remain always in the finite? For every finite index > > > > both numbers are different. > > > > > > But when you use all finite indices they are the same. But the indices > > > are still finite. And by the axiom of infinity we can use the set of > > > all finite indices. > > > > But not in sum{i = 1..n} i = (n + 1) * n / 2 ? > > That formula is about a single natural number n, it is *not* about a set > of natural numbers. The left hand side is obviously about a set of natural numbers. > > > > Sorry, that makes it unreadable to me. But from what I can decipher is > > > that they define the infinite sum of cardinal numbers as the cardinality > > > of the infinite union of disjoint sets. In common set theory the infinite > > > union of sets is defined, and so this makes sense. But this has nothing > > > to do with > > > sum{i = 1..n} i = (n + 1) * n /2 > > > where the sum is constrained to natural numbers n. Analysis and algebra > > > are about real numbers, and in those fields infinite sums are not defined, > > > except as a limit. Moreover, using their definition of limits of > > > sequences of ordinal numbers we would get omega. It can make sense to > > > make > > > sum{i = 1..oo} i = aleph_0 > > > but it does not follow in any way from the usual theorem, and requires > > > some additional definitions. > > > > No. Then sum{i = 1 to oo} i could also yield another result. But > > that is certaily false. It can neither be finite nor can it be > > uncountable. The same holds for sum{i = 1 to oo} 1 = aleph_0. > > Why? That depends entirely on how you define it. I have looked, what you > are missing is that in the book cardinal arithmetic is defined, and that is > not the same as standard arithmetic, and with cardinal arithmetic the use > of 'oo' is extremely misleading, and actually also your notation. Better is > (and I think the writers of the book use that notation): > sum{i in N} i = aleph_0, > which is quite different because it does not suggest sequencing. The writers sum the *sequence* 1, 2, 3, ... with a definite and unique result (no definition could lead to another result. Your claim that this was not done or was impossible is wrong. > And indeed > also sum{i in S} i is defined when S is an arbitrary set of cardinal numbers. > But indeed, the book *gives* some additional definitions to get such results. > In addition, it is also stated that those additional definitions require the > axiom of choice. Did you really read it? In standard arithmetic > sum{i = 1..oo} i is not defined. Here the sum *is* defined. You said it was not, but it is. > > In cardinal arithmetic division (and subtraction, I think) are not defined. > And I clearly stated that the formula was about natural numbers, so I think > I implied sufficiently that I meant ordinary arithmetic. I said that the sequence of natural numbers can be summed up. This is true. > But whatever the > case when I use their definition of the limit of sequences of ordinal > numbers, I can define: > sum{i = 1..omega} i = lim{n -> omega} sum{1 = 1..n} i = > lim{n -> omega} (n + 1) * n / 2 = omega. omega and aleph are not distinct in this case. Ordinal and cardinal summation are not different in this case. Hrbacek and Jech show that 1 + 2 + 3 + ... = aleph_0 (you can also call it omega). And in fact it cannot be else. But the formula which is true for all natural numbers and for the sum of every initial segment of natural numbers is not true for every initial segment of natural numbers. Regards, WM
From: Virgil on 17 May 2007 16:28 In article <1179427901.710765.103690(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > I agree with the common mathematical definition Wm does not agree with any common mathematical definition if he can help it. He does not seem to like being pinned down to only one unambiguous meaning. It leaves him so little room to maneuver in.
From: Virgil on 17 May 2007 16:38
In article <1179433414.192576.40870(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Mai, 02:23, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1179346082.257799.291...(a)k79g2000hse.googlegroups.com> WM > > If something is stated to hold for all natural numbers, that means that > > it holds for each and every natural number, *not* that it holds for the > > set of natural numbers. > > > If a sum is stated to hold for every subset of natural numbers then it > has to hold for the whole set too. It is not stated to hold for EVERY subset, it does not, for instance, hold for the set of even naturals. There are certain subsets of N called initial sets, which consist of the set of naturals bounded above by some natural, and the sum is only defined for such sets. And N is not such a set outside WM's MathUnRealism. > > > > > > But not in sum{i = 1..n} i = (n + 1) * n / 2 ? > > > > That formula is about a single natural number n, it is *not* about a set > > of natural numbers. > > The left hand side is obviously about a set of natural numbers. Does WM claim that N is a set of that form, i.e., the set of naturals bounded above by some explicit natural n? If so then WMe should be able to produce that natural. So what is it, WM? |