Cantor Confusion
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From: William Hughes on 12 Oct 2006 13:59 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > > > > 0.1 > > > > > > > > 0.11 > > > > > > > > 0.111 > > > > > > > > ... > > > > > > > > > > > > > > > > > > > > That is correct. But every element of the natural numbers is finite. > > > > > > > Hence every element covers its predecessors. If 0.111... is covered by > > > > > > > "the whole list", then it is covered by one element. That, however, is > > > > > > > excuded. > > > > > > > > > > > > > > > > > > > Since no one has claimed that '0.111... is covered by "the whole > > > > > > list"', I fail > > > > > > to see the relevence of a sentence that starts out > > > > > > 'If 0.111... is covered by "the whole list"'. > > > > > > > > > > If every digit position is well defined, then 0.111... is covered "up > > > > > to every position" by the list numbers, which are simply the natural > > > > > indizes. I claim that covering "up to every" implies covering "every". > > > > > > > > > Quantifier dyslexia. > > > > > > Quantifier magic may apply and may be useful at several occasions. But > > > to state that in a unary representation of natural numbers the union of > > > "up to every" and "every" have different meaning is easily disproved. > > > > > > > The fact that > > > > > > > > for every digit position N, there exists a natural number, M, > > > > such that M covers 0.111... to position N > > > > > > > > does not imply > > > > > > > > there exist a natural number M such that for every digit > > > > position N, M covers 0.111... to position N > > > > > > In case of linear sets we have a third statement which is true without > > > doubt: > > > > > > 3) Every set of unary numbers which covers 0.111... to a finite > > > position N can be replaced by a single unary number. > > > > > > > > This holds for every finite position N. If 0.111... has only finite > > > positions, then (3) holds for every position. As 0.111... does not > > > consist of mre than every position, it holds for the whole number > > > 0.111.... > > > > > > > So we have > > > > for every digit position N, there exists a set of > > unary numbers which covers 0.111... to position > > N > > > > and > > > > for every digit position N, if there exists a set of unary > > numbers which covers 0.111... to position N, > > there exists a single unary number, M, > > such that M covers 0.111... to position N > > > > this implies > > > > for every digit position N, > > there exists a single unary number, M, > > such that M covers 0.111... to position N > > > > > > this does not imply > > > > there exists a single unary number M such that for every digit > > position N, M covers 0.111... to position N > > Why shouldn't it? Because for every digit position N, there exists a single unary number, M, such that M covers 0.111... to position N can be true even if M depends on N. The statement there exists a single unary number M such that for every digit position N, M covers 0.111... to position N cannot be true if M depends on N. Therefore the first does not imply the second. >If every digit position of 0.111... is a finite > position then exactly this is implied. No, the fact that every digit position of 0.111... is finite does not mean that M does not depend on N, and that is what we need. What we need is not that "every digit position of 0.111... is finite" but "there are a finite number of digit positions in 0.111..." These two statements are not the same. - William Hughes
From: mueckenh on 12 Oct 2006 14:01 Dik T. Winter schrieb: > In article <1160650371.242557.284430(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > So the definition I gave for a limit of a sequence of sets you agree > > > > > with? Or not? I am seriously confused. With the definition I gave, > > > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > > > > > Sorry, I don't understand your definition. > > > > > > What part of the definition do you not understand? I will repeat it here: > > > > What *might* be a sensible definition of a limit for a sequence of sets of > > > > naturals is, that (given each A_n is a set of naturals), the limit > > > > lim{n = 1 ... oo} A_n = A > > > > exists if and only if for every p in N, there is an n0, such that either > > > > (1) p in A_n for n > n0 > > > > or > > > > (2) p !in A_n for n > n0. > > > > In the first case p is in A, in the second case p !in A. > > > Pray, read the complete definition before you give comments. > > > > I do not believe that definition (2) is of any relevance. > > It is. > > > Cantor uses Lim{n} n = omega witout much ado. > > That is not a limit of "sets". It is the limit of the natural numbers. The limit is omega, an ordinal number. Meanwhile we know that every number is a set. Hence, it is a limit of sets. > > > In his first paper he uses even Wallis' symbol oo. What should there > > require a definition, if all natural did exist? > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > = N. > > Yes, and that fits my definition. On the other hand, how would you > define lim{n --> oo} {n, n+1, ...}? I would not attempt to define that. > > > Not this expression is meaningless but the assumption behind it, the > > complete set N. > > In your not so humble opinion. In reality. Regards, WM
From: William Hughes on 12 Oct 2006 14:06 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not be > > > > > a largest. > > > > > > > > There is no such thing as a number with arbitrary size > > > > (i.e. a number that has the property that its size is > > > > arbitrary). > > > > > > The number I will write down here has that property, before I write it > > > down. > > > > > > 7 > > > > > > Now it has no longer this property. > > > > > > > > > The prime number to be discovered next has the property to be unknown. > > > - Until it is discovered. > > > > You are confusing "unknown" with "arbitrary". They do > > not mean the same thing. > > I understand by "arbitrary" just what you understand: to be determined > at will. Yes, the term arbitrary describes the method of choice. It does not describe the number chosen. > If I write down "7" so it was arbitrary, just my choice to do > so. Choices can be arbitrary, so yes you can make an arbitrary choice. Numbers cannot be arbitrary, so the the fact that you happened to choose "7" does not make 7 arbitrary. > > > > > > > > > > > > Take a finite set B. > > > > > > > > By definition, B must have a finite > > > > number of elements. This > > > > number does not have arbitrary size. > > > > > > > > Each element has a size This size is not > > > > arbitrary. > > > > > > > > So we have a finite number (which does not > > > > have arbitrary size) > > > > of elements, each of which does not > > > > have arbitrary size. > > > > > > > > Thus B has a largest element. > > > > > > What is the largest number of the set of numbers mentioned in the New > > > York Times in 2007? > > > > "unknown" but not "arbitrary". > > correct, unless I pay an advertisment posting that number. > > > > > You see, your proof is rubbish. B will have a largest element. And the > > > set of all numbers ever used in the universe in eternity also will have > > > a largest element. But it has not yet. > > > > Therefore it is unknown. However, it is not arbitrary. > > The largest element possible with 100 bits can be very different, > according to my arbitrary choice of representation. > Note again, it is your method of choice, not the thing that you choose, which can be arbitrary or not arbitrary. - William Hughes
From: David Marcus on 12 Oct 2006 14:09 Dik T. Winter wrote: > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > To inform the set theorist about the possible existence of sets with > > finite cardinality but without a largest number. > > Interesting but in contradiction with the definition of the concept of > "finite set". So you are talking about something else than "finite > sets". It would seem he is. I don't understand why people use words in non- standard ways without explaining what they mean. They are guaranteeing that no one will understand them. -- David Marcus
From: William Hughes on 12 Oct 2006 14:13
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > It is not > > > contradictory to say that in a finite set of numbers there need not be > > > a largest. > > > > It contradicts the definition of "finite set". But I know that you are > > not interested in definitions. > > We know that a set of numbers consisting altogether of 100 bits cannot > contain more than 100 numbers. Therefore the set is finite. The largest > number of such a set cannot be determined, as far as I know. There is a big difference between saying we do not know what the value of the largest element of a set is and saying that a set does not have a largest element. - William Hughes |