Cantor Confusion
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From: mueckenh on 9 Feb 2007 04:04 On 8 Feb., 13:48, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1170853587.867792.9...(a)v33g2000cwv.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 7 Feb., 02:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1170761231.100893.322...(a)v33g2000cwv.googlegroups.com> mueck.= > ....@rz.fh-augsburg.de writes: > > > > > > > > On 5 Feb., 05:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > > > > What *is* IIII. You never have defined it. You really do not like > > > > > definitions, as they pin down the real meaning. > > > > > > > > IIII is a primitive. Everybody knows it - even without definition by > > > > Peano or Dedekind. That means, we do not need Peano to know small > > > > natural numbers. > > > > > > Perhaps not. But in axiomatic mathematics it is best to define it. > > > Unless you want to do non-axiomatic mathematics, like Cantor. The > > > problem with non-axiomatic mathematics is that proofs are so difficult > > > as nothing is defined and people can talk at cross-purposes. It is > > > quite possible that I have quite different ideas about IIII than you > > > have. And indeed, that is the case. I do not see it as the number 4, > > > but as a sequence of four strokes. In my opinion the successor to > > > that would be four strokes with a fifth stroke starting at the bottom > > > left and ending at the top right. > > > > Why not. IIIII can be abbreviated by 5 or V or your proposal or ... > > Again, still no definition. So IIIIII can be abbreviated 5I, I5, VI or > IV? And IIIIIIIIII can be abbreviated 55 or VV? That is a matter of convention. Therefore the unary represantation, which is not a matter of convention, is preferable. > > > > > > When I asked you about what basic > > > > > way, III c IV c V, you answered that I had to continue with IIII, > > > > > IIIII, etc. > > > > > > > > The basic way to establish IV c V is to use the numbers in their basic > > > > form IIII c IIIII. (Numbers *are* their representations.) > > > > > > No. Numbers are abstract entities. Their representations are > > > concretisations of those abstract entities. > > > > Where and what are these entities? > > Abstract entities. Where are they? Further your answer does not distinguish between numbers and impressions of colour or beauty. If you say a number exists, then you must be able to distinguish it from *everything* else (including other numbers. This is certainly not done by saying: abstract entity. > > > > are many different representations possible, and they may even be > > > contradictionary when you mix representations. If you see the greek > > > letter "delta" do you associate that immediately with the number 4? > > > And if you see the letter "lambda" do you associate it with the number > > > 30? Nevertheless, they *are* representations. > > > > You have to use some agrrement if you want to use abbreviations. For > > IIII you need no agreement. > > Oh. I think one is needed. I believe that without any other cultural contact two persons could start communicating by unary numbers. > > > > > > > Why do you say N is wrong in (2) but not in (1)? > > > > > > > > > > Where in (1) is N? I do not see N at all. > > > > > > > > "1 ist eine nat=FCrliche Zahl" means "1 in N". > > > > > > But N is not yet defined in (1). You give meaning beyond what is stated. > > > > If you use "natürliche Zahl" or "N" does not matter. You use something > > not yet defined (and never defined by Peano). > > You apparently do not know how a recursive definition works. (1), (2) > (when properly corrected) define the natural numbers. (3) defines the > set of natural numbers. I know it. Cantor already used it (see p. 128 of my book). That does not mean that it is good. (1), (2) have not to be corrected (they appear in several text books). Your assertion that " x is in N" and "x is a natural number" were different is wrong. Both has to be defined. > > > > > The property "being a > > > > natural number" implies the existence of N. > > > > > > How can that be the case if N is not yet mentioned or even defined? > > > > N is only an abbrevation of set of natural numbers. > > An undefined abbreviation. When doing definitions you should do it the > correct way. See: > We are going to define natural numbers: > (1) 1 is a natural number > (2) if a is a natural number, the successor of a is also a natural > number > These two together define the natural numbers. Not yet. According to your definition also -7 and pi could be natural numbers. But we could write your two axioms also as: > We are going to define N: > (1) 1 is in N > (2) if a is in N, the successor of a is also in N. > These two together define N. > > > Not yet, I am in chapter 9 now. But that is a ridiculous definition, as > > > that is extremely circular. And with that definition you will not be > > > able to show that 2 is a subset of 3. > > > > IT IS NOT A DEFINITION. It is the criterion of existence. > > So when I ask for a definition you refuse to give a definition? I gave two definitions. Peano and that with +1 which I would attribute to Dedekind. > > > > Correct. There is no infinite line existing. All we assume is a line > > longer than any line we have measured yet. > > Such lines also do not have physical existence. Each and every physical line > has a width smaller than anything measured yet. And I do not think that > physical lines are really straight either. The physical existence of a line is "a measurable distance" between two points. The points exists as sets of coordinates. Regards, WM
From: mueckenh on 9 Feb 2007 04:15 On 8 Feb., 13:37, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 8, 3:07 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > On 7 Feb., 16:02, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > On Feb 7, 9:29 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > For all for finite natural numbers > > > > > > n = |{2,4,6,...,2n}| < 2n > > > > > > Now take the limit as n-->oo (the < becomes <= in a limit, > > > > > No. n < 2n is always true for natural numbers (which property excludes > > > > 0). There is no reason to assume that 1/2 becomes 1 in "the limit". > > > > > > lim[n-->oo] n = oo, lim[n-->oo] 2n = oo) > > > > > > oo <= oo > > > > > > There is no contradiction. > > > > > Wrong for all finite numbers. > > > > And since the limit is not a finite number > > > Every number which can appear in this inequality is a finite number. > > If something can appear in place of n which is not a finite number, > > then you agree that N contains infinite numbers. > > No. The thing that may be able to appear in this inequality > is the cardinality of E. Either > > -The cardinality of E does not exist > or > -The cardiality of E is not a finite number. > > In neither case does N contain infinite numbers. > In neither case is the cardinality of E a finite number. > In neither case is there a contradiction. You can refrain from repeating over and over again your dogmas. In potential infinity there is no cardinal number E of an infinite set, because there are only finite sets. > > >But that is wrong for > > any set N, be it actually or potentially infinite. > > Since I do no make such a claim, this statment > is irrelevent. > > > > the fact that there > > > is a contradiction for all finite numbers does not mean that > > > there is a contradiction for the limit. > > > Potentially infinite means here only that the value represented by n > > can become as large as we like. > > Recall the statment we are discussing is > > -E contains numbers which are larger > than the cardinal number of > > You note: "Potentially infinite means here only that the value > represented by n > can become as large as we like" > > It follows immediately that either > > -The cardinality of E does not exist > or > -The cardiality of E is not a finite number. > > The fact that potentially infinite means that the set E > can become as large as we like is very relevent. In any case what we like must be finite because we cannot like wha cannot exist. > > > > > Can you answer the following question yes or no? > > > > > > Is the potentially infinite set of finite even numbers > > > > > a set of finite even numbers? > > > > > Yes, I can. The answer is yes. Probably set theory gives another > > > > answer. > > > > Let E be the potentially infinite set of finite even numbers > > > > You have now made three claims > > > > -every set of finite even numbers contains numbers which > > > are larger than the cardinal number of the set > > > True. > > > > - E is a set of finite even numbers > > > Yes, but not a complete or finished set as set theory requires. > > > > - the statment "E contains numbers which are larger > > > than the cardinal number of the E" is false > > > No. The answer is not so simple. > > The statement can be proved correct for an actually existing set E > > with a cardinal number |E| which is a number that can be compared by > > size with natural numbers. > > The statement is false for a potentially infinite set E because such > > sets do not have cardinal numbers. > > In other words the statement > > A: every set of finite even numbers contains numbers which > are larger than the cardinal number of the set > > is false. No, it is true. How can you conclude it was wrong? I said: > > The statement can be proved correct for an actually existing set E > > with a cardinal number |E| which is a number that can be compared by > > size with natural numbers. > The statement > > B: every set of finite even numbers with a finite cardinal > contains numbers which are larger than the cardinal > number of the set > > The statement B is true, and of little interest. You are misinterpreting again. Correct it: > B: every set of finite even numbers > contains numbers which are larger than the cardinal > number of the set. This implies that the cardinal nunmber of the set is finite. > As has been pointed out before, if you wish to claim > that only sets with finite cardinality exist, then > you are free to do so. I do not wish it, but the finity of numbers enforces it. > > However, saying that E does not have a cardinal > does not change the properties of E. E still > has a sparrow (recall a sparrow is an equivalence > class under the equivalence relation equitransform > which generalizes the concept of bijection to include > potentially infinite sets). This sparrow can be > compared with other sparrow's including the sparrows > of finite sets, which are just the finite cardinals. > There is nothing contradictory about defining > a sparrow. Unless you say that it is a number larger than any natural number. > > Extending the concept of cardinality to include > potentially infinite sets does not lead to > a contradiction. Unless you say that it is a number larger than any natural number. Regards, WM
From: mueckenh on 9 Feb 2007 04:20 On 8 Feb., 14:10, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 8, 2:42 am, mueck...(a)rz.fh-augsburg.de wrote: > > > On 7 Feb., 14:35, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > The potentially infinite set of even numbers is *constructed* by its > > > > segments > > > > > {2,4,6,...,2n} > > > > > Every time we increase n by 1 we increase 2n by 2. This cannot be > > > > avoided. Therefore it is impossible to have for finite natural numbers > > > > > lim[n-->oo] |{2,4,6,...,2n}| > 2n > > > > The limit of finite natural numbers is not a finite natural number. > > > There is no limit. > > And since something that does not exist is not a finite > natural number the statement > > -The limit of finite natural numbers is not a finite natural > number. > > is true. I agree. Alas, this sentence looks very political and could establish the "fact" that the limit exists. > > > The notation lim[n-->oo] says nothing but that > > there is *no* limit. The size of n is not bounded. > > One can of course choose whether or not > lim[n-->oo] n exists. However, whether or not it > exists it will not be a finite natural number. > > Statements that are true for all finite natural numbers > may or may not be true for things that are > not finite natural numbers. Very wise. But the fact that some statements about some entities are not true does not imply the existence of these entities in some esoteric form. Regards, WM
From: mueckenh on 9 Feb 2007 04:28 On 9 Feb., 00:50, Virgil <vir...(a)comcast.net> wrote: > In article <1170934078.553109.258...(a)s48g2000cws.googlegroups.com>, > > mueck...(a)rz.fh-augsburg.de wrote: > > On 8 Feb., 11:02, Franziska Neugebauer <Franziska- > > Neugeba...(a)neugeb.dnsalias.net> wrote: > > > > This is not true. A tree *then* is the structure plus the set of paths > > > and/or nodes. You persistently try to vaporize the essential > > > constituents of a tree until it fits your preconception. > > > What is wrong with my use of trees, except that it is inconvenient if > > one is a bit clumsy in adapting new ideas? > > Among other things, it hides essential properties within naming rules > for the nodes. Use only: Set of nodes which belong to a tree or path or level. Write tree or path or level in oder to be conceise. > > > > But we need not use this order. > > > > To show that you are wrong even in this picture we *need* this order: > > > But we don't need it for our conclusion. Therefore we do not use this > > picture. Show what is wrong with the conclusion that the nodes of a > > path are a subset of the set of nodes of the tree. > > Is every ordered subset of nodes of a tree a path in that tree? No. > > If not, WM's analysis is flawed. No. > > Given an ordered subset which is a path, is every other ordering of that > set also a path? > Forget about ordering. The paths exist without our conversation abou them. Consider sets of nodes only. Regards, WM
From: mueckenh on 9 Feb 2007 04:30
On 9 Feb., 08:41, "Ross A. Finlayson" <r...(a)tiki-lounge.com> wrote: > Hi Wolfgang, > > I think you, too, should support having sci.math.infinity, and instead > post your topics there, or on sci.math as I don't care. > > Please do so. Thank you, I am about to show that infinity does not exist. So I will not create sci.math.infinity. Regards, WM |