An uncountable countable set
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From: Virgil on 23 Jul 2006 14:17 In article <1153674750.702920.61130(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > The numbers count the lines. As long as there are only finite numbers, > > > the list is finite. > > > > Why? There are infinitely many finite numbers. > > Just that is in question. More precisely: Are there actually or only > potentially infinitely many numbers. With respect to what axiom system are you asking this? Or do you maintain that there is some sort of absolute truth in mathematics that does not require any assumptions at all? > > > You are trying to establish > > that the Axiom of Infinity leads to a contradiction. But your very > > statement > > "As long as there are only finite numbers, the list is finite" is in itself > > in contradiction to the axiom. The statement is just opinion, without > > proof. > > So there is not yet a contradiction. Please show a proof of your > > statement. > > The proof is given by the list: > > 1 0.1 > 2 0.11 > 3 0.111 > ... ... > w 0.111... That is not a list as presented. > > The set of unary numbers cannot result in the *+sum 0.111... . (I > defined: the *+ sum is 1 if there is at least one 1 in it.) With > respect to the left column this means the cardinality of 1,2,3,... > cannot be w without w in it. Since "mueckenh"'s definition of "*+"is in all cases cited equivalent to taking the LUB of the corresponding set of ordinals, one finds that "mueckenh" is, as so often before, still wrong. At least in ZF and NBG. > Again you forgot the special from of unary representations. Is the > special property of unary representation so difficult to see, so hard > to remember? Properties of sets of numbers are not dependent on any representational scheme.
From: Virgil on 23 Jul 2006 14:21 In article <1153674914.649461.71180(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1153478871.303029.5360(a)i3g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > But it is obvious for *all* *finite* *unary* number. And only finite > > > unary numbers are involved in the second list below: > > > > > > The *- sum of > > > > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > 0.111... > > > > 0.111... is not a finite unary number. Rather, I would say it is not a > > unary number at all. > > 0.111.. is not a terminating rational number. Rather I would say it is > not (a representation of) a rational number at all. If 0.111... neither a unary nor a rational, and"meuckenh"'s "*+" is only defined for unary or rationals, then 0.111... is not defined at all as a result from the "*+" operation.
From: Virgil on 23 Jul 2006 14:32 In article <1153675669.168444.262950(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > [reformatting to standard discussion style] > > > > In article <1153479321.840071.258130(a)b28g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter wrote: > > > > You again refrain from answering questions. The definition of > > > > 0.999... > > > > is as follows: > > > > 0.999... = lim{n -> oo} sum{k = 1 .. n} 10^(-k) > > > > You said that definition is silly. What is silly about it? > > > > > > Sorry, could you help me with "silly". My find function shows with > > > "silly" only the sentence: "It is only in modern set theory that such > > > silly things as the rejection of the binary tree occur." > > > > You actually said that it made no sense. But I will try to find the > > correct quote. > > I confirm: It makes no sense because omega makes no > sense. It makes good sense in NBG. What is "mueckenh"'s axiom system in which it makes no sense? > > Every edge can be interpreted as a binary digit, 0 if it goes to the > left, and 1, if it goes to the right, for instance. Thus every edge is identified with a terminating sequence of of binary digits, but every path identifies, by a similar correspondence, to a non-terminating sequence of binary digits. So that "mueckenh" says that the cardinality of the set of terminating sequences equals that of the set of non-terminating sequences. Which is foolish of him. > > Just this is the mess in set theory. Infinity is not present in > magnitudes of numbers. It is most important to distinguish this. It is not one of the natural numbers, but no finite natural number can be the count of all natural numbers.
From: Virgil on 23 Jul 2006 14:36 In article <1153678443.193262.86440(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Each column is treated separately. Therefore the *+sum does not depend > on the number of columns. It depends directly on which columns have 1's in them, which requires directly that one know "how many" columns have 1's in them.
From: Dik T. Winter on 23 Jul 2006 21:09
In article <1153674750.702920.61130(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > The numbers count the lines. As long as there are only finite numbers, > > > the list is finite. > > > > Why? There are infinitely many finite numbers. > > Just that is in question. More precisely: Are there actually or only > potentially infinitely many numbers. With the axiom of infinity there are infinitely many finite numbers. And I still do not know the exact difference between actual infinite and potential infinite. But that is (in my opinion) something that is best left to philosphy. > > You are trying to establish > > that the Axiom of Infinity leads to a contradiction. But your very > > statement "As long as there are only finite numbers, the list is finite" > > is in itself in contradiction to the axiom. The statement is just > > opinion, without proof. So there is not yet a contradiction. Please > > show a proof of your statement. > > The proof is given by the list: > > 1 0.1 > 2 0.11 > 3 0.111 > ... ... > w 0.111... That is neither a proof, nor a list. > The set of unary numbers cannot result in the *+sum 0.111... . (I > defined: the *+ sum is 1 if there is at least one 1 in it.) With > respect to the left column this means the cardinality of 1,2,3,... > cannot be w without w in it. You again state such without a proof, prove it, assuming that there is no last element. Let's start with the list: A1 = 0.1 A2 = 0.11 A3 = 0.111 assuming decimal notation (because in that way we can define everything we need in the easiest possible way). Now it is easy to define *+ on two operands, Ap *+ Aq = Ap when p > q else it is Aq. And now it is easy to do a finite set of operations, and we can proof that: (*+){i = 1 .. n} Ai = An no problem so far. We get a problem when we try to define the operation for *all* Ai. As there are not finitely many Ai, we can not use the finite case to extend to all elements of the list. In my opinion the only sane way to define the *+ sum of all elements is to define it as lim{n -> oo} An but you apparently disagree. Whatever, with this definition we get that (*+){i = 1 .. oo} Ai = 1/9 and 1/9 is not equal to any of the Ai. > > > I never said so. > > > > So, why do you think that it is in the list, when the list is a list of > > natural numbers in unary notation? > > It is not *in* the list, but has been added at that position which is > due to its 1's, namely omega. This is justified because all numbers are > listed at that position which is due to the 1's in their unary > representations and because omega is a number (though not a natural) > which covers only *one* place in the sequence of ordinals. Except that there is no digit position with ordinal number w in the standard decimal notation. > > But they give different information. When the "notations" are unary > > representations of natural numbers, their *+ sum is not the representation > > of natural numbers, and so is not in the list. When the "notations" > > are decimal (or ternary, or whatever) representations of rational numbers > > their *+ sum is a representation of a rational number that is not in the > > list. > > So in both cases, the *+ sum is not in the list. In both cases the *+ > sum is a representation of a number which is different from all list > numbers (naturals or terminating rationals). Yes, indeed. > > In both cases, when you insert the *+ sum early in the list you get either > > (case 1) a non-natural number in the list, so it makes no sense, or > > (case 2) a different rational in the list, and in this case the *+ sum > > is that rational. But in the latter case I see no problem. > > In the list there are only terminating rationals. 0.111... has omega > digits. Therefore we see perfect parallels. Again, you make no sense here. I think you are arguing (case 2). If you add the *+ sum in the list the list no longer contains only terminating rationals. Apparently you are thinking that 0.111... is a terminating rational. What is the ordinal number of its last digit (and please mind that in decimal notation the ordinal numbers of the digits are natural numbers)? > > > This is only possible, if it was in the list before already. > > > > Wrong. It is because all elements are idempotent under your *+ operation. > > Union {a} with {b} is the same as union {a} with {b} and {a, b}. > > Again you forgot the special from of unary representations. Is the > special property of unary representation so difficult to see, so hard > to remember? And again switching back to unary representation. In unary representation 0.111... is *not* a natural number. As the list contains natural numbers only, that "number" is not in the list. However, what I said still holds. Seeing the things 0.1, etc. as strings of digits, and the *+ operation as an idempotent operation on those strings, it becomes quite clear that given a set of strings, and calculating the *+ sum, that inclusion that sum in the set does not change the *+ sum. It is not necessary to have that *+ sum already in the set before doing the summing. So your "this is only possible, ..." makes no sense. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |