An uncountable countable set
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From: Han de Bruijn on 25 Sep 2006 07:40 MoeBlee wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>MoeBlee wrote: >> >>>Han de Bruijn wrote: >>> >>>>to mainstream mathematics and its illusionary "rigour". >>> >>>The rigor is in a recursive axiomatization with recursive rules of >>>inference. You haven't shown that this is an illusion. >> >>That rigor turns out to be incompatible with useful infinitesimals. > > You think the rigor itself is incompatible? Absolute rigor is a phantom. > Why don't your read a book on mathematical logic? I have read dozens of them. Doesn't help. On the contrary ... Han de Bruijn
From: mueckenh on 25 Sep 2006 07:57 Dik T. Winter schrieb: > In article <1158313360.862711.19070(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > That does not exist. At least it is not described by n. III is a > > > > representation of 3. > > > > > > Now you shift back to representation, pray remain consistent. > > > > Representation is number. There is no difference. Numerals have no > > "soul". > > Makes no sense to me. You may represent a natural number as you wish, > that does not change anything. Correct. 3 or III or {D,i,k} is a means which can be used to calculate with. All those objects are numbers which can be used by an intelligent being to count. Of course, also 3.000 is a number and "three", and perhaps some intelligent being uses "kashdgaejh" to denote the number three. There is some relativity. > > > > > If you know the positions by heart, then you need no addition actually. > > > > You had already used it or the person who devised that technique had > > > > used it. But earlier or later your knowing by heart will end and you > > > > will have to count +1. > > > > > > Nope. I will only to need to know successors. Anyhow, you read much more > > > in the successor of the Peano axioms than is present. The successor is > > > defined without even any knowledge of addition at all. So > > > succ(George V) = Edward VIII, succ(Edward VIII) = George VI and > > > succ(George VI) = Elizabeth II within the set of British kings and queens. > > > I do not think what way of addition you would propose for that. Of > > > course, this successor function does not satisfy all of Peano's axioms, > > > but I hope you get the idea. > > > > I was talking of *counting* (remember: Zahl and zaehlen) which requires > > natural numbers which require the ability to add 1 after you have run > > out of the numbers known by heart. > > You stated that you needed counting to determine the successor. That is > false. The successor is defined without any reference to counting. The successor function *is* counting (+1). The successors are defined without counting only over a very restricted domain. In the usual decimal systems only from 1 to 12 and then repeating again and again from X to X + 9. > > > > > The old Greek and othe cultures have just used their letters as numbers > > > > too. > > > > > > Yes, every culture had their representation of numbers. Sometimes base 10, > > > sometimes other bases. Base 20 is quite common in Europe. Mixed bases > > > are also used. Sometimes they were used to count from 1. Sometimes from > > > 0. > > > > Who did so before Cantor? > > Some people from India. There are quite a few where you can read about > the zeroth year of the reign of somebody. Have a look at the vast number > of different calendars and year countings that have been in use in India. That is curious, but consequent. Our calendar consists of years BC and AC but does not contain a year zero. That frequently leads to errors. Of course it is correct, to count divisible objects (like years or electric current) beginning with 0, 0.1, 0.333 or so. But that is not applicable to counting of indvisible elements of sets. Either you have none or you have more, i.e., at least one. Hence, there may be a first element, but not a zeroth. Regards, WM
From: mueckenh on 25 Sep 2006 08:21 Dik T. Winter schrieb: > In article <1158313932.806912.250350(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1158159208.048232.260050(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <45070bcb(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > > > ... > > > > > > What about an unboundedly large but finite number? > > > > > > > > > > Let me ask you to, before I can answer such a question. What is your > > > > > definition of "number"? (I asked the same from Wolfgang Mueckenheim, > > > > > but his answer was not satisfactory, also not to himself, I think, > > > > > because he never answered to questions about it.) > > > > > > > > Didn't you read my paper on the physical constraints of numbers? > > > > > > Yes, I read it. Mathematically it makes no sense. > > > > You did not yet recognize it, perhaps later. But you were not telling > > the truth above, were you? > > Where did I not write the truth? Here: >>>>What is your > > > > definition of "number"? (I asked the same from Wolfgang Mueckenheim, > > > > but his answer was not satisfactory, also not to himself, I think, > > > > because he never answered to questions about it.) > > > If I considered Dik as one person, I would write {Dik, Virgil, me} for > > instance. But you are right, there is some ambiguity. > > Yes, so it is not a proper definition. And as such, the above to me still > makes no sense as a proper definition at all. There is a natural number which is the largest one ever mentioned or thought during the lifetime o the universe. It is not properly defined before the universe ceases and probably also not afterwards. But it is or will be (depending on the question of determination or not). Nevertheless, it does not exist yet. We have to live with those improper objects. > > > > > 2) or is completely determined by a series of digits. > > > > > > Question. A terminating or a non-terminating series? > > > > There are only terminating series. There is no infinity in reality and > > useful mathematics. > > Oh. So you state. But 1/3 is a number? 1/3 is a number, properly defined, for instance, by the pair of numbers 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you cannot index all positions, you cannot distinguish the positions of this number from those with finite sequences (and you cannot distinguish them from other infinte sequences which could exist, if one could exist). You cannot localize the digit number [pi*10^100]. But this problem does not matter, because every digit is the same, 3. Therefore I am not sure about the numerical status of 0.333... . In the binary system 1/3 is not well defined. You cannot know whether the digit number [pi*10^100] is 0 or 1. > > What do you mean with "exisiting"? Existing is a thing you can use, like the largest known Fermat-prime, or the theorem of Pythagoras, or a hot dog in your hands. > The set of prime numbers is infinite > and unbounded. The set of known prime numbers is finite and bounded. it is finite, but not bound, because it can, and probably will, grow. That is the same as with the set of natural or real numbers. Regards, WM
From: MoeBlee on 25 Sep 2006 08:44 Han de Bruijn wrote: > Absolute rigor is a phantom. There are recursively axiomatized theories. You deem that not to be absolute rigor. What are your reasons? > > Why don't your read a book on mathematical logic? > > I have read dozens of them. Doesn't help. On the contrary ... Would you give an example of a text that you read and why didn't help? Maybe we can find the source of some misunderstanding you have. MoeBlee
From: Han de Bruijn on 25 Sep 2006 10:41
MoeBlee wrote: > Han de Bruijn wrote: > >>Absolute rigor is a phantom. > > There are recursively axiomatized theories. You deem that not to be > absolute rigor. What are your reasons? Absolute also means that e.g. an Alien Civilization can understand those axiomatized theories. What makes you think that such will be the case? >>>Why don't your read a book on mathematical logic? >> >>I have read dozens of them. Doesn't help. On the contrary ... > > Would you give an example of a text that you read and why didn't help? > Maybe we can find the source of some misunderstanding you have. They are all in Dutch. But maybe I'll take a look at it tonight. Han de Bruijn |